Find the exact values of the six trigonometric functions of .

1.

SOLUTION:

The length of the side opposite is 8 , the length of the side adjacent to is 14, and the length of the hypotenuseis 18.

2.

SOLUTION:

The length of the side opposite is 2 , the length of the side adjacent to is 13, and the length of the hypotenuse is 15.

3.

SOLUTION:

The length of the side opposite is 9, the length of the side adjacent to is 4, and the length of the hypotenuse is

.

4.

SOLUTION:

The length of the side opposite is 12, the length of the side adjacent to is 35, and the length of the hypotenuse is 37.

5.

SOLUTION:

The length of the side opposite is , the length of the side adjacent to is 26, and the length of the hypotenuse is 29.

6.

SOLUTION:

The length of the side opposite is 30, the length of the side adjacent to is 5 , and the length of the hypotenuse is 35.

7.

SOLUTION:

The length of the side opposite is 6 and the length of the hypotenuse is 10. By the Pythagorean Theorem, the

length of the side adjacent to is = or8.

8.

SOLUTION:

The length of the side opposite is 8 and the length of the side adjacent to is 32. By the Pythagorean Theorem, the length of the hypotenuse is

= or8 .

Use the given trigonometric function value of the acute angle to find the exact values of the five remaining trigonometric function values of .

9.sin =

SOLUTION:

Draw a right triangle and label one acute angle . Because sin = = , label the opposite side 4 and the

hypotenuse 5.

By the Pythagorean Theorem, the length of the side adjacent to is or3.

10.cos =

SOLUTION:

Draw a right triangle and label one acute angle . Because cos = = , label the adjacent side 6 and the

hypotenuse 7.

By the Pythagorean Theorem, the length of the side opposite is or .

11.tan =3

SOLUTION:

Draw a right triangle and label one acute angle . Because tan = =3 or , label the side opposite 3 and

the adjacent side 1.

By the Pythagorean Theorem, the length of the hypotenuse is or .

12.sec =8

SOLUTION:

Draw a right triangle and label one acute angle . Because sec = =8or , label the hypotenuse 8 and

the adjacent side 1.

By the Pythagorean Theorem, the length of the side adjacent to is = or3 .

13.cos =

SOLUTION:

Draw a right triangle and label one acute angle . Because cos = = , label the adjacent side 5 and the

hypotenuse 9.

By the Pythagorean Theorem, the length of the side opposite is or2 .

14.tan =

SOLUTION:

Draw a right triangle and label one acute angle . Because tan = = , label the side opposite 5 and

adjacent side 4.

By the Pythagorean Theorem, the length of the hypotenuse is or .

15.cot =5

SOLUTION:

Draw a right triangle and label one acute angle . Because cot = =5 or ,labelthesideadjacentto as

5 and the opposite side 1.

By the Pythagorean Theorem, the length of the hypotenuse is or .

16.csc =6

SOLUTION:

Draw a right triangle and label one acute angle . Because csc = =6 or , label the hypotenuse 6 and the

side opposite as 1.

By the Pythagorean Theorem, the length of the adjacent side is or .

17.sec =

SOLUTION:

Draw a right triangle and label one acute angle . Because sec = = , label the hypotenuse 9 and the side

opposite as 2.

By the Pythagorean Theorem, the length of the opposite side is or .

18.sin =

SOLUTION:

Draw a right triangle and label one acute angle . Because sin = = , label the side opposite as 8 and

the hypotenuse 13.

By the Pythagorean Theorem, the length of the adjacent side is or .

Find the value of x. Round to the nearest tenth.

19.

SOLUTION:An acute angle measure and the length of the hypotenuse are given, so the sine function can be used to find the

length of the side opposite .

20.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the tangent function can be used to find the length of the side adjacent to .

21.

SOLUTION:An acute angle measure and the length of the hypotenuse are given, so the cosine function can be used to find the

length of the side adjacent to .

22.

SOLUTION:

An acute angle measure and the length of the side adjacent to are given, so the cosine function can be used to find the length of the hypotenuse.

23.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the sine function can be used to find thelength of the hypotenuse.

24.

SOLUTION:

An acute angle measure and the length of the side adjacent to are given, so the tangent function can be used to find the length of the side opposite .

25.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the tangent function can be used to find the length of the side adjacent to .

26.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the sine function can be used to find thelength of the hypotenuse.

27.MOUNTAIN CLIMBINGAteamofclimbersmustdeterminethewidthofaravineinordertosetupequipment to cross it. If the climbers walk 25 feet along the ravine from their crossing point, and sight the crossing point on the far side of the ravine to be at a 35 angle, how wide is the ravine?

SOLUTION:

An acute angle measure and the adjacent side length are given, so the tangent function can be used to find the length of the opposite side.

Therefore, the ravine is about 17.5 feet wide.

28.SNOWBOARDINGBradbuiltasnowboardingrampwithaheightof3.5feetandan18 incline. a. Draw a diagram to represent the situation. b. Determine the length of the ramp.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 18 and the opposite side 3.5 feet.

b. Because an acute angle measure and opposite side length are given, the sine function can be used to find the length of the hypotenuse.

Therefore, the length of the ramp is about 11.3 feet.

29.DETOURTrafficisdetouredfromElwoodAve.,left0.8mileonMapleSt.,andthenrightonOakSt.,whichintersectsElwoodAve.ata32angle. a. Draw a diagram to represent the situation. b. Determine the length of Elwood Ave. that is detoured.

SOLUTION:a. Draw a right triangle with acute angle 32 and opposite side length 0.8 mile. Label the side opposite the 32 angleMaple St., the hypotenuse Oak St., and the adjacent side Elwood Ave.

b. Because an acute angle and the length of opposite side are given, the tangent function can be used to find the adjacent side length.

Therefore, the length of Elwood Ave. that is detoured is about 1.3 miles.

30.PARACHUTINGAparatrooperencountersstrongerwindsthananticipatedwhileparachutingfrom1350feet,causing him to drift at an 8 angle. How far from the drop zone will the paratrooper land?

SOLUTION:Because an acute angle and the length of the side that is adjacent to the angle are given, the tangent function can be used to find the length of the opposite side.

So, the paratrooper will land about 190 feet away from the drop zone.

Find the measure of angle . Round to the nearest degree, if necessary.

31.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

32.

SOLUTION:

Because the length of the hypotenuse and side adjacent to are given, the cosine function can be used to find .

33.

SOLUTION:

Because the length of the hypotenuse and side opposite are given, the sine function can be used to find .

34.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

35.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

36.

SOLUTION:

Because the length of the hypotenuse and side adjacent to are given, the cosine function can be used to find .

37.

SOLUTION:

Because the length of the hypotenuse and side opposite are given, the sine function can be used to find .

38.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

39.PARASAILING Kayladecidedtotryparasailing.Shewasstrappedintoaparachutetowedbyaboat.An800-foot line connected her parachute to the boat, which was at a 32 angle of depression below her. How high above the water was Kayla?

SOLUTION:The angle of elevation from the boat to the parachute is equivalent to the angle of depression from the parachute tothe boat because the two angles are alternate interior angles, as shown below.

Because an acute angle and the hypotenuse are given, the sine function can be used to find x.

Therefore, Kayla was about 424 feet above the water.

40.OBSERVATION WHEEL TheLondonEyeisa135-meter-tall observation wheel. If a passenger at the top of the wheel sights the London Aquarium at a 58 angle of depression, what is the distance between the aquarium andthe London Eye?

SOLUTION:The angle of depression from the top of the wheel to the aquarium is 58, which means that the angle of elevation from the aquarium to the top of the wheel is also 58 because they are alternate interior angles. Draw a diagram of a right triangle and label one acute angle 58 and the opposite side 135 m.

Because an acute angle and the side length opposite the angle are given, the tangent function can be used to find x.

Therefore, the distance between the aquarium and the London Eye is about 84 meters.

41.ROLLER COASTER Onarollercoaster,375feetoftrackascendata55 angle of elevation to the top before the first and highest drop. a. Draw a diagram to represent the situation. b. Determine the height of the roller coaster.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 55 and the hypotenuse 375 feet.

b. Because an acute angle and the length of the hypotenuse are given, you can use the sine function to find the length of the opposite side.

Therefore, the height of the roller coaster is about 307 feet.

42.SKI LIFTAcompanyisinstallinganewskiliftona225-meter-high mountain that will ascend at a 48 angle of elevation. a. Draw a diagram to represent the situation. b. Determine the length of cable the lift requires to extend from the base to the peak of the mountain.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 48 and the opposite side 225 meters.

b. Because an acute angle and the length of the side opposite the angle are given, you can use the sine function to find the length of the hypotenuse.

So, the company will need about 303 meters of cable.

43.BASKETBALLBothDerekandSamare5feet10inchestall.Dereklooksata10-foot basketball goal with an angleofelevationof29,andSamlooksatthegoalwithanangleofelevationof43.IfSamisdirectlyinfrontofDerek, how far apart are the boys standing?

SOLUTION:Draw a diagram to model the situation. The vertical distance from the boys' heads to the rim is 10(12) [5(12) + 10] or 50 inches. Label the horizontal distance between Sam and Derek as x and the horizontal distance between Sam and the goal as y .

From the smaller right triangle, you can use the tangent function to find y .

From the larger right triangle, you can use the tangent function to find x.

Therefore, Derek and Sam are standing about 36.6 inches or 3.1 feet apart.

44.PARISAtouristonthefirstobservationleveloftheEiffelTowersightstheMuseDOrsay at a 1.4 angle of depression.Atouristonthethirdobservationlevel,located219metersdirectlyabovethefirst,sightstheMuseDOrsay at a 6.8 angle of depression. a. Draw a diagram to represent the situation. b. DeterminethedistancebetweentheEiffelTowerandtheMuseDOrsay.

SOLUTION:

a.

b. Because the angles of depression from the 1st

and 3rdlevelstotheMuseDOrsay are 1.4 and 6.8,

respectively,theanglesofelevationfromtheMuseDOrsaytothe1st and 3rd levels are also 1.4 and 6.8, respectively.

Use the tangent function to write an equation for the smaller right triangle in terms of y .

Next, use the tangent function to write an equation for the larger right triangle in terms of y .

Set the equations that you found for each triangle equal to one another and solve for x.

Therefore,thedistancebetweentheEiffelTowerandtheMuseDOrsay is about 2310 meters.

45.LIGHTHOUSETwoshipsarespottedfromthetopofa156-foot lighthouse. The first ship is at a 27 angle of depression, and the second ship is directly behind the first at a 7 angle of depression. a. Draw a diagram to represent the situation. b. Determine the distance between the two ships.

SOLUTION:a.

b. From the smaller right triangle, you can use the tangent function to find y .

From the larger right triangle, you can use the tangent fuction to find x.

Therefore, the distance between the two ships is about 964 feet.

46.MOUNT RUSHMOREThefacesofthepresidentsatMountRushmoreare60feettall.Avisitorseesthetopof George Washingtons head at a 48 angle of elevation and his chin at a 44.76 angle of elevation. Find the height of Mount Rushmore.

SOLUTION:From the smaller triangle, you can use the tangent function to find y .

From the larger triangle, you can use the tangent function to find y , too.

Next, set the two equations equal to one another to solve for x.

Therefore, Mount Rushmore is about 500 feet tall.

Solve each triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

47.

SOLUTION:Use trigonometric functions to find b and c.

Because the measures of two angles are given, B can be found by subtracting A from

Therefore, b 16.5,c 17.5.

48.

SOLUTION:Use trigonometric functions to find y and z .

Because the measures of two angles are given, X can be found by subtracting Z from

Therefore, y 37.1,z 32.5.

49.

SOLUTION:Use the Pythagorean Theorem to find r.

Use the tangent function to find P.

Because the measures of two angles are now known, you can find Q by subtracting P from

Therefore, P 43, Q 47, and r 34.0.

50.

SOLUTION:Use the Pythagorean Theorem to find d.

Use the cosine function to find D.

Because the measures of two angles are now known, you can find E by subtracting D from

Therefore, D 77, E 13, and d 29.2.

51.

SOLUTION:Use trigonometric functions to find j and k .

Because the measures of two angles are given, K can be found by subtracting J from

Therefore, j 18.0,k 6.2.

52.

SOLUTION:Use the Pythagorean Theorem to find y .

Use the sine function to find W.

Because the measures of two angles are now known, you can find Y by subtracting W from

Therefore, and y 3.9.

53.

SOLUTION:Use trigonometric functions to find f and h.

Because the measures of two angles are given, H can be found by subtracting F from

Therefore, f 19.6,h 17.1.

54.

SOLUTION:Use the Pythagorean Theorem to find t.

Use the tangent function to find R.

Because the measures of two angles are now known, you can find S by subtracting R from

Therefore, and t 8.1.

55.BASEBALL Michaels seat at a game is 65 feet behind home plate. His line of vision is 10 feet above the field. a. Draw a diagram to represent the situation. b. What is the angle of depression to home plate?

SOLUTION:a.

b. Michaels angle of depression to home plate is equivalent to the angle of elevation from home plate to where he is sitting.

Use the tangent function to find .

Therefore, the angle of depression to home plate is about

56.HIKING Jessica is standing 2 miles from the center of the base of Pikes Peak, and looking at the summit of the mountain, which is 1.4 miles from the base. a. Draw a diagram to represent the situation. b. With what angle of elevation is Jessica looking at the summit of the mountain?

SOLUTION:a.

b. Use the tangent function to find the angle of elevation.

Therefore, the angle of elevation is about

Find the exact value of each expression without using a calculator.57.sin 60

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side opposite the 60 angle is x and the length of the hypotenuse is 2x.

58.cot30

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side adjacent to the 30 angle is x and the length of the opposite side is x.

59.sec30

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the hypotenuse is 2x and the length of the side adjacent to the 30 angle is x.

60.cos45

SOLUTION:Draw a diagram of a 45-45-90 triangle.

The side length is x and the hypotenuse is x.

61.tan60

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side opposite the 60 is x and the length of the adjacent side is x.

62.csc45

SOLUTION:Draw a diagram of a 45-45-90 triangle.

The length of the hypotenuse is x and the side length is x.

Without using a calculator, find the measure of the acute angle that satisfies the given equation.63.tan = 1

SOLUTION:

Because tan = 1 and tan = , it follows that =1.Inthe triangle below, the side length

opposite an acute angle is 1 and the adjacent side length is also 1. So, =1.

Therefore, = 45.

64.cos =

SOLUTION:

Because cos = and cos = , it follows that = . In the triangle below, the side

length adjacent to the 30 angle is and the hypotenuse is 2. So, = .

Therefore, = 30.

65.cot =

SOLUTION:

Because cot = and cot = , it follows that = . In the triangle below, the side

length that is adjacent to the 60 angle is 1 and the length of the opposite side is . So, = or .

Therefore, = 60.

66.sin =

SOLUTION:

Because sin = and sin = , it follows that = . In the triangle below, the side

length opposite an acute angle is 1 and the hypotenuse is .So, = or .

Therefore, =45.

67.csc = 2

SOLUTION:

Because csc = 2 and csc = , it follows that =2or . In the triangle below, the

hypotenuse is 2 and the side length that is opposite the 30 angle is 1. So, = or2.

Therefore, =30.

68.sec = 2

SOLUTION:

Because sec = 2 and sec = , it follows that =2or . In the triangle below, the

hypotenuse is 2 and the side length that is adjacent to the 60 angle is 1. So, = or2.

Therefore, = 60.

Without using a calculator, determine the value of x.

69.

SOLUTION:Because an acute angle, the hypotenuse, and the length of the side adjacent to the angle are given, the cosine

function can be used to find x. Using the properties of triangles, you can find that cos 30 = .

70.

SOLUTION:

Because the triangle is a triangle, the legs are the same length.

71.SCUBA DIVINGAscubadiverlocated20feetbelowthesurfaceofthewaterspotsashipwreckata70 angle of depression. After descending to a point 45 feet above the ocean floor, the diver sees the shipwreck at a 57 angle of depression. Draw a diagram to represent the situation, and determine the depth of the shipwreck.

SOLUTION:First, draw a diagram to represent the situation.

Label the horizontal distance from the shipwreck to the point on the ocean floor below the diver as x. Label the vertical distance from the point 20 feet below sea level to the point 45 feet below sea level as y . Find the complementary angle for each angle of depression. Use the tangent function to write an equation for the smaller right triangle in terms of x.

Use the tangent function to write an equation for the larger right triangle in terms of x.

Set the two equations equal to one another to solve for y .

Therefore, the depth of the shipwreck is 20 + 35 + 45 or about 100 feet.

Find the value of cos if is the measure of the smallest angle in each type of right triangle.72.3-4-5

SOLUTION:Draw a diagram of a 3-4-5 triangle. The smallest angle will be the angle opposite the side with a length of 3.

73.5-12-13

SOLUTION:Draw a diagram of a 5-12-13 triangle. The smallest angle will be the angle opposite the side with a length of 5.

74.SOLAR POWERFindthetotalareaofthepanelshownbelow.

SOLUTION:

The area of the panel is given by , where the width is 10 feet and the length is unknown. Notice that the length of the panel is also the hypotenuse of a right triangle with an acute angle of 55 and an opposite side length of 3.5 feet. The sine function can be used to find the hypotenuse of the triangle.

So, the length of the panel is 4.27 feet. Find the area.

Therefore, the area of the panel is 42.7 square feet.

Without using a calculator, insert the appropriate symbol >, cot 60.

76.tan60 cot30

SOLUTION:

Use a graphing calculator to find tan 60 and cot 30. To find cot 30, find .

tan 60 1.732 cot 30 1.732 Therefore,tan60=cot30.

77.cos30 csc45

SOLUTION:

Use a graphing calculator to find cos 30 and csc 45. To find csc 45, find .

cos 30 0.866 csc 45 1.414 Therefore,cos30 csc60.

80.tan45 sec30

SOLUTION:tan45sec30

Use a graphing calculator to find tan 45 and sec 30. To find sec 30, find .

tan 45 = 1 sec 30 1.155 Therefore,tan45

Find the exact values of the six trigonometric functions of .

1.

SOLUTION:

The length of the side opposite is 8 , the length of the side adjacent to is 14, and the length of the hypotenuseis 18.

2.

SOLUTION:

The length of the side opposite is 2 , the length of the side adjacent to is 13, and the length of the hypotenuse is 15.

3.

SOLUTION:

The length of the side opposite is 9, the length of the side adjacent to is 4, and the length of the hypotenuse is

.

4.

SOLUTION:

The length of the side opposite is 12, the length of the side adjacent to is 35, and the length of the hypotenuse is 37.

5.

SOLUTION:

The length of the side opposite is , the length of the side adjacent to is 26, and the length of the hypotenuse is 29.

6.

SOLUTION:

The length of the side opposite is 30, the length of the side adjacent to is 5 , and the length of the hypotenuse is 35.

7.

SOLUTION:

The length of the side opposite is 6 and the length of the hypotenuse is 10. By the Pythagorean Theorem, the

length of the side adjacent to is = or8.

8.

SOLUTION:

The length of the side opposite is 8 and the length of the side adjacent to is 32. By the Pythagorean Theorem, the length of the hypotenuse is

= or8 .

Use the given trigonometric function value of the acute angle to find the exact values of the five remaining trigonometric function values of .

9.sin =

SOLUTION:

Draw a right triangle and label one acute angle . Because sin = = , label the opposite side 4 and the

hypotenuse 5.

By the Pythagorean Theorem, the length of the side adjacent to is or3.

10.cos =

SOLUTION:

Draw a right triangle and label one acute angle . Because cos = = , label the adjacent side 6 and the

hypotenuse 7.

By the Pythagorean Theorem, the length of the side opposite is or .

11.tan =3

SOLUTION:

Draw a right triangle and label one acute angle . Because tan = =3 or , label the side opposite 3 and

the adjacent side 1.

By the Pythagorean Theorem, the length of the hypotenuse is or .

12.sec =8

SOLUTION:

Draw a right triangle and label one acute angle . Because sec = =8or , label the hypotenuse 8 and

the adjacent side 1.

By the Pythagorean Theorem, the length of the side adjacent to is = or3 .

13.cos =

SOLUTION:

Draw a right triangle and label one acute angle . Because cos = = , label the adjacent side 5 and the

hypotenuse 9.

By the Pythagorean Theorem, the length of the side opposite is or2 .

14.tan =

SOLUTION:

Draw a right triangle and label one acute angle . Because tan = = , label the side opposite 5 and

adjacent side 4.

By the Pythagorean Theorem, the length of the hypotenuse is or .

15.cot =5

SOLUTION:

Draw a right triangle and label one acute angle . Because cot = =5 or ,labelthesideadjacentto as

5 and the opposite side 1.

By the Pythagorean Theorem, the length of the hypotenuse is or .

16.csc =6

SOLUTION:

Draw a right triangle and label one acute angle . Because csc = =6 or , label the hypotenuse 6 and the

side opposite as 1.

By the Pythagorean Theorem, the length of the adjacent side is or .

17.sec =

SOLUTION:

Draw a right triangle and label one acute angle . Because sec = = , label the hypotenuse 9 and the side

opposite as 2.

By the Pythagorean Theorem, the length of the opposite side is or .

18.sin =

SOLUTION:

Draw a right triangle and label one acute angle . Because sin = = , label the side opposite as 8 and

the hypotenuse 13.

By the Pythagorean Theorem, the length of the adjacent side is or .

Find the value of x. Round to the nearest tenth.

19.

SOLUTION:An acute angle measure and the length of the hypotenuse are given, so the sine function can be used to find the

length of the side opposite .

20.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the tangent function can be used to find the length of the side adjacent to .

21.

SOLUTION:An acute angle measure and the length of the hypotenuse are given, so the cosine function can be used to find the

length of the side adjacent to .

22.

SOLUTION:

An acute angle measure and the length of the side adjacent to are given, so the cosine function can be used to find the length of the hypotenuse.

23.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the sine function can be used to find thelength of the hypotenuse.

24.

SOLUTION:

An acute angle measure and the length of the side adjacent to are given, so the tangent function can be used to find the length of the side opposite .

25.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the tangent function can be used to find the length of the side adjacent to .

26.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the sine function can be used to find thelength of the hypotenuse.

27.MOUNTAIN CLIMBINGAteamofclimbersmustdeterminethewidthofaravineinordertosetupequipment to cross it. If the climbers walk 25 feet along the ravine from their crossing point, and sight the crossing point on the far side of the ravine to be at a 35 angle, how wide is the ravine?

SOLUTION:

An acute angle measure and the adjacent side length are given, so the tangent function can be used to find the length of the opposite side.

Therefore, the ravine is about 17.5 feet wide.

28.SNOWBOARDINGBradbuiltasnowboardingrampwithaheightof3.5feetandan18 incline. a. Draw a diagram to represent the situation. b. Determine the length of the ramp.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 18 and the opposite side 3.5 feet.

b. Because an acute angle measure and opposite side length are given, the sine function can be used to find the length of the hypotenuse.

Therefore, the length of the ramp is about 11.3 feet.

29.DETOURTrafficisdetouredfromElwoodAve.,left0.8mileonMapleSt.,andthenrightonOakSt.,whichintersectsElwoodAve.ata32angle. a. Draw a diagram to represent the situation. b. Determine the length of Elwood Ave. that is detoured.

SOLUTION:a. Draw a right triangle with acute angle 32 and opposite side length 0.8 mile. Label the side opposite the 32 angleMaple St., the hypotenuse Oak St., and the adjacent side Elwood Ave.

b. Because an acute angle and the length of opposite side are given, the tangent function can be used to find the adjacent side length.

Therefore, the length of Elwood Ave. that is detoured is about 1.3 miles.

30.PARACHUTINGAparatrooperencountersstrongerwindsthananticipatedwhileparachutingfrom1350feet,causing him to drift at an 8 angle. How far from the drop zone will the paratrooper land?

SOLUTION:Because an acute angle and the length of the side that is adjacent to the angle are given, the tangent function can be used to find the length of the opposite side.

So, the paratrooper will land about 190 feet away from the drop zone.

Find the measure of angle . Round to the nearest degree, if necessary.

31.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

32.

SOLUTION:

Because the length of the hypotenuse and side adjacent to are given, the cosine function can be used to find .

33.

SOLUTION:

Because the length of the hypotenuse and side opposite are given, the sine function can be used to find .

34.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

35.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

36.

SOLUTION:

Because the length of the hypotenuse and side adjacent to are given, the cosine function can be used to find .

37.

SOLUTION:

Because the length of the hypotenuse and side opposite are given, the sine function can be used to find .

38.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

39.PARASAILING Kayladecidedtotryparasailing.Shewasstrappedintoaparachutetowedbyaboat.An800-foot line connected her parachute to the boat, which was at a 32 angle of depression below her. How high above the water was Kayla?

SOLUTION:The angle of elevation from the boat to the parachute is equivalent to the angle of depression from the parachute tothe boat because the two angles are alternate interior angles, as shown below.

Because an acute angle and the hypotenuse are given, the sine function can be used to find x.

Therefore, Kayla was about 424 feet above the water.

40.OBSERVATION WHEEL TheLondonEyeisa135-meter-tall observation wheel. If a passenger at the top of the wheel sights the London Aquarium at a 58 angle of depression, what is the distance between the aquarium andthe London Eye?

SOLUTION:The angle of depression from the top of the wheel to the aquarium is 58, which means that the angle of elevation from the aquarium to the top of the wheel is also 58 because they are alternate interior angles. Draw a diagram of a right triangle and label one acute angle 58 and the opposite side 135 m.

Because an acute angle and the side length opposite the angle are given, the tangent function can be used to find x.

Therefore, the distance between the aquarium and the London Eye is about 84 meters.

41.ROLLER COASTER Onarollercoaster,375feetoftrackascendata55 angle of elevation to the top before the first and highest drop. a. Draw a diagram to represent the situation. b. Determine the height of the roller coaster.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 55 and the hypotenuse 375 feet.

b. Because an acute angle and the length of the hypotenuse are given, you can use the sine function to find the length of the opposite side.

Therefore, the height of the roller coaster is about 307 feet.

42.SKI LIFTAcompanyisinstallinganewskiliftona225-meter-high mountain that will ascend at a 48 angle of elevation. a. Draw a diagram to represent the situation. b. Determine the length of cable the lift requires to extend from the base to the peak of the mountain.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 48 and the opposite side 225 meters.

b. Because an acute angle and the length of the side opposite the angle are given, you can use the sine function to find the length of the hypotenuse.

So, the company will need about 303 meters of cable.

43.BASKETBALLBothDerekandSamare5feet10inchestall.Dereklooksata10-foot basketball goal with an angleofelevationof29,andSamlooksatthegoalwithanangleofelevationof43.IfSamisdirectlyinfrontofDerek, how far apart are the boys standing?

SOLUTION:Draw a diagram to model the situation. The vertical distance from the boys' heads to the rim is 10(12) [5(12) + 10] or 50 inches. Label the horizontal distance between Sam and Derek as x and the horizontal distance between Sam and the goal as y .

From the smaller right triangle, you can use the tangent function to find y .

From the larger right triangle, you can use the tangent function to find x.

Therefore, Derek and Sam are standing about 36.6 inches or 3.1 feet apart.

44.PARISAtouristonthefirstobservationleveloftheEiffelTowersightstheMuseDOrsay at a 1.4 angle of depression.Atouristonthethirdobservationlevel,located219metersdirectlyabovethefirst,sightstheMuseDOrsay at a 6.8 angle of depression. a. Draw a diagram to represent the situation. b. DeterminethedistancebetweentheEiffelTowerandtheMuseDOrsay.

SOLUTION:

a.

b. Because the angles of depression from the 1st

and 3rdlevelstotheMuseDOrsay are 1.4 and 6.8,

respectively,theanglesofelevationfromtheMuseDOrsaytothe1st and 3rd levels are also 1.4 and 6.8, respectively.

Use the tangent function to write an equation for the smaller right triangle in terms of y .

Next, use the tangent function to write an equation for the larger right triangle in terms of y .

Set the equations that you found for each triangle equal to one another and solve for x.

Therefore,thedistancebetweentheEiffelTowerandtheMuseDOrsay is about 2310 meters.

45.LIGHTHOUSETwoshipsarespottedfromthetopofa156-foot lighthouse. The first ship is at a 27 angle of depression, and the second ship is directly behind the first at a 7 angle of depression. a. Draw a diagram to represent the situation. b. Determine the distance between the two ships.

SOLUTION:a.

b. From the smaller right triangle, you can use the tangent function to find y .

From the larger right triangle, you can use the tangent fuction to find x.

Therefore, the distance between the two ships is about 964 feet.

46.MOUNT RUSHMOREThefacesofthepresidentsatMountRushmoreare60feettall.Avisitorseesthetopof George Washingtons head at a 48 angle of elevation and his chin at a 44.76 angle of elevation. Find the height of Mount Rushmore.

SOLUTION:From the smaller triangle, you can use the tangent function to find y .

From the larger triangle, you can use the tangent function to find y , too.

Next, set the two equations equal to one another to solve for x.

Therefore, Mount Rushmore is about 500 feet tall.

Solve each triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

47.

SOLUTION:Use trigonometric functions to find b and c.

Because the measures of two angles are given, B can be found by subtracting A from

Therefore, b 16.5,c 17.5.

48.

SOLUTION:Use trigonometric functions to find y and z .

Because the measures of two angles are given, X can be found by subtracting Z from

Therefore, y 37.1,z 32.5.

49.

SOLUTION:Use the Pythagorean Theorem to find r.

Use the tangent function to find P.

Because the measures of two angles are now known, you can find Q by subtracting P from

Therefore, P 43, Q 47, and r 34.0.

50.

SOLUTION:Use the Pythagorean Theorem to find d.

Use the cosine function to find D.

Because the measures of two angles are now known, you can find E by subtracting D from

Therefore, D 77, E 13, and d 29.2.

51.

SOLUTION:Use trigonometric functions to find j and k .

Because the measures of two angles are given, K can be found by subtracting J from

Therefore, j 18.0,k 6.2.

52.

SOLUTION:Use the Pythagorean Theorem to find y .

Use the sine function to find W.

Because the measures of two angles are now known, you can find Y by subtracting W from

Therefore, and y 3.9.

53.

SOLUTION:Use trigonometric functions to find f and h.

Because the measures of two angles are given, H can be found by subtracting F from

Therefore, f 19.6,h 17.1.

54.

SOLUTION:Use the Pythagorean Theorem to find t.

Use the tangent function to find R.

Because the measures of two angles are now known, you can find S by subtracting R from

Therefore, and t 8.1.

55.BASEBALL Michaels seat at a game is 65 feet behind home plate. His line of vision is 10 feet above the field. a. Draw a diagram to represent the situation. b. What is the angle of depression to home plate?

SOLUTION:a.

b. Michaels angle of depression to home plate is equivalent to the angle of elevation from home plate to where he is sitting.

Use the tangent function to find .

Therefore, the angle of depression to home plate is about

56.HIKING Jessica is standing 2 miles from the center of the base of Pikes Peak, and looking at the summit of the mountain, which is 1.4 miles from the base. a. Draw a diagram to represent the situation. b. With what angle of elevation is Jessica looking at the summit of the mountain?

SOLUTION:a.

b. Use the tangent function to find the angle of elevation.

Therefore, the angle of elevation is about

Find the exact value of each expression without using a calculator.57.sin 60

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side opposite the 60 angle is x and the length of the hypotenuse is 2x.

58.cot30

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side adjacent to the 30 angle is x and the length of the opposite side is x.

59.sec30

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the hypotenuse is 2x and the length of the side adjacent to the 30 angle is x.

60.cos45

SOLUTION:Draw a diagram of a 45-45-90 triangle.

The side length is x and the hypotenuse is x.

61.tan60

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side opposite the 60 is x and the length of the adjacent side is x.

62.csc45

SOLUTION:Draw a diagram of a 45-45-90 triangle.

The length of the hypotenuse is x and the side length is x.

Without using a calculator, find the measure of the acute angle that satisfies the given equation.63.tan = 1

SOLUTION:

Because tan = 1 and tan = , it follows that =1.Inthe triangle below, the side length

opposite an acute angle is 1 and the adjacent side length is also 1. So, =1.

Therefore, = 45.

64.cos =

SOLUTION:

Because cos = and cos = , it follows that = . In the triangle below, the side

length adjacent to the 30 angle is and the hypotenuse is 2. So, = .

Therefore, = 30.

65.cot =

SOLUTION:

Because cot = and cot = , it follows that = . In the triangle below, the side

length that is adjacent to the 60 angle is 1 and the length of the opposite side is . So, = or .

Therefore, = 60.

66.sin =

SOLUTION:

Because sin = and sin = , it follows that = . In the triangle below, the side

length opposite an acute angle is 1 and the hypotenuse is .So, = or .

Therefore, =45.

67.csc = 2

SOLUTION:

Because csc = 2 and csc = , it follows that =2or . In the triangle below, the

hypotenuse is 2 and the side length that is opposite the 30 angle is 1. So, = or2.

Therefore, =30.

68.sec = 2

SOLUTION:

Because sec = 2 and sec = , it follows that =2or . In the triangle below, the

hypotenuse is 2 and the side length that is adjacent to the 60 angle is 1. So, = or2.

Therefore, = 60.

Without using a calculator, determine the value of x.

69.

SOLUTION:Because an acute angle, the hypotenuse, and the length of the side adjacent to the angle are given, the cosine

function can be used to find x. Using the properties of triangles, you can find that cos 30 = .

70.

SOLUTION:

Because the triangle is a triangle, the legs are the same length.

71.SCUBA DIVINGAscubadiverlocated20feetbelowthesurfaceofthewaterspotsashipwreckata70 angle of depression. After descending to a point 45 feet above the ocean floor, the diver sees the shipwreck at a 57 angle of depression. Draw a diagram to represent the situation, and determine the depth of the shipwreck.

SOLUTION:First, draw a diagram to represent the situation.

Label the horizontal distance from the shipwreck to the point on the ocean floor below the diver as x. Label the vertical distance from the point 20 feet below sea level to the point 45 feet below sea level as y . Find the complementary angle for each angle of depression. Use the tangent function to write an equation for the smaller right triangle in terms of x.

Use the tangent function to write an equation for the larger right triangle in terms of x.

Set the two equations equal to one another to solve for y .

Therefore, the depth of the shipwreck is 20 + 35 + 45 or about 100 feet.

Find the value of cos if is the measure of the smallest angle in each type of right triangle.72.3-4-5

SOLUTION:Draw a diagram of a 3-4-5 triangle. The smallest angle will be the angle opposite the side with a length of 3.

73.5-12-13

SOLUTION:Draw a diagram of a 5-12-13 triangle. The smallest angle will be the angle opposite the side with a length of 5.

74.SOLAR POWERFindthetotalareaofthepanelshownbelow.

SOLUTION:

The area of the panel is given by , where the width is 10 feet and the length is unknown. Notice that the length of the panel is also the hypotenuse of a right triangle with an acute angle of 55 and an opposite side length of 3.5 feet. The sine function can be used to find the hypotenuse of the triangle.

So, the length of the panel is 4.27 feet. Find the area.

Therefore, the area of the panel is 42.7 square feet.

Without using a calculator, insert the appropriate symbol >, cot 60.

76.tan60 cot30

SOLUTION:

Use a graphing calculator to find tan 60 and cot 30. To find cot 30, find .

tan 60 1.732 cot 30 1.732 Therefore,tan60=cot30.

77.cos30 csc45

SOLUTION:

Use a graphing calculator to find cos 30 and csc 45. To find csc 45, find .

cos 30 0.866 csc 45 1.414 Therefore,cos30 csc60.

80.tan45 sec30

SOLUTION:tan45sec30

Use a graphing calculator to find tan 45 and sec 30. To find sec 30, find .

tan 45 = 1 sec 30 1.155 Therefore,tan45

Find the exact values of the six trigonometric functions of .

1.

SOLUTION:

The length of the side opposite is 8 , the length of the side adjacent to is 14, and the length of the hypotenuseis 18.

2.

SOLUTION:

The length of the side opposite is 2 , the length of the side adjacent to is 13, and the length of the hypotenuse is 15.

3.

SOLUTION:

The length of the side opposite is 9, the length of the side adjacent to is 4, and the length of the hypotenuse is

.

4.

SOLUTION:

The length of the side opposite is 12, the length of the side adjacent to is 35, and the length of the hypotenuse is 37.

5.

SOLUTION:

The length of the side opposite is , the length of the side adjacent to is 26, and the length of the hypotenuse is 29.

6.

SOLUTION:

The length of the side opposite is 30, the length of the side adjacent to is 5 , and the length of the hypotenuse is 35.

7.

SOLUTION:

The length of the side opposite is 6 and the length of the hypotenuse is 10. By the Pythagorean Theorem, the

length of the side adjacent to is = or8.

8.

SOLUTION:

The length of the side opposite is 8 and the length of the side adjacent to is 32. By the Pythagorean Theorem, the length of the hypotenuse is

= or8 .

Use the given trigonometric function value of the acute angle to find the exact values of the five remaining trigonometric function values of .

9.sin =

SOLUTION:

Draw a right triangle and label one acute angle . Because sin = = , label the opposite side 4 and the

hypotenuse 5.

By the Pythagorean Theorem, the length of the side adjacent to is or3.

10.cos =

SOLUTION:

Draw a right triangle and label one acute angle . Because cos = = , label the adjacent side 6 and the

hypotenuse 7.

By the Pythagorean Theorem, the length of the side opposite is or .

11.tan =3

SOLUTION:

Draw a right triangle and label one acute angle . Because tan = =3 or , label the side opposite 3 and

the adjacent side 1.

By the Pythagorean Theorem, the length of the hypotenuse is or .

12.sec =8

SOLUTION:

Draw a right triangle and label one acute angle . Because sec = =8or , label the hypotenuse 8 and

the adjacent side 1.

By the Pythagorean Theorem, the length of the side adjacent to is = or3 .

13.cos =

SOLUTION:

Draw a right triangle and label one acute angle . Because cos = = , label the adjacent side 5 and the

hypotenuse 9.

By the Pythagorean Theorem, the length of the side opposite is or2 .

14.tan =

SOLUTION:

Draw a right triangle and label one acute angle . Because tan = = , label the side opposite 5 and

adjacent side 4.

By the Pythagorean Theorem, the length of the hypotenuse is or .

15.cot =5

SOLUTION:

Draw a right triangle and label one acute angle . Because cot = =5 or ,labelthesideadjacentto as

5 and the opposite side 1.

By the Pythagorean Theorem, the length of the hypotenuse is or .

16.csc =6

SOLUTION:

Draw a right triangle and label one acute angle . Because csc = =6 or , label the hypotenuse 6 and the

side opposite as 1.

By the Pythagorean Theorem, the length of the adjacent side is or .

17.sec =

SOLUTION:

Draw a right triangle and label one acute angle . Because sec = = , label the hypotenuse 9 and the side

opposite as 2.

By the Pythagorean Theorem, the length of the opposite side is or .

18.sin =

SOLUTION:

Draw a right triangle and label one acute angle . Because sin = = , label the side opposite as 8 and

the hypotenuse 13.

By the Pythagorean Theorem, the length of the adjacent side is or .

Find the value of x. Round to the nearest tenth.

19.

SOLUTION:An acute angle measure and the length of the hypotenuse are given, so the sine function can be used to find the

length of the side opposite .

20.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the tangent function can be used to find the length of the side adjacent to .

21.

SOLUTION:An acute angle measure and the length of the hypotenuse are given, so the cosine function can be used to find the

length of the side adjacent to .

22.

SOLUTION:

An acute angle measure and the length of the side adjacent to are given, so the cosine function can be used to find the length of the hypotenuse.

23.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the sine function can be used to find thelength of the hypotenuse.

24.

SOLUTION:

An acute angle measure and the length of the side adjacent to are given, so the tangent function can be used to find the length of the side opposite .

25.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the tangent function can be used to find the length of the side adjacent to .

26.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the sine function can be used to find thelength of the hypotenuse.

27.MOUNTAIN CLIMBINGAteamofclimbersmustdeterminethewidthofaravineinordertosetupequipment to cross it. If the climbers walk 25 feet along the ravine from their crossing point, and sight the crossing point on the far side of the ravine to be at a 35 angle, how wide is the ravine?

SOLUTION:

An acute angle measure and the adjacent side length are given, so the tangent function can be used to find the length of the opposite side.

Therefore, the ravine is about 17.5 feet wide.

28.SNOWBOARDINGBradbuiltasnowboardingrampwithaheightof3.5feetandan18 incline. a. Draw a diagram to represent the situation. b. Determine the length of the ramp.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 18 and the opposite side 3.5 feet.

b. Because an acute angle measure and opposite side length are given, the sine function can be used to find the length of the hypotenuse.

Therefore, the length of the ramp is about 11.3 feet.

29.DETOURTrafficisdetouredfromElwoodAve.,left0.8mileonMapleSt.,andthenrightonOakSt.,whichintersectsElwoodAve.ata32angle. a. Draw a diagram to represent the situation. b. Determine the length of Elwood Ave. that is detoured.

SOLUTION:a. Draw a right triangle with acute angle 32 and opposite side length 0.8 mile. Label the side opposite the 32 angleMaple St., the hypotenuse Oak St., and the adjacent side Elwood Ave.

b. Because an acute angle and the length of opposite side are given, the tangent function can be used to find the adjacent side length.

Therefore, the length of Elwood Ave. that is detoured is about 1.3 miles.

30.PARACHUTINGAparatrooperencountersstrongerwindsthananticipatedwhileparachutingfrom1350feet,causing him to drift at an 8 angle. How far from the drop zone will the paratrooper land?

SOLUTION:Because an acute angle and the length of the side that is adjacent to the angle are given, the tangent function can be used to find the length of the opposite side.

So, the paratrooper will land about 190 feet away from the drop zone.

Find the measure of angle . Round to the nearest degree, if necessary.

31.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

32.

SOLUTION:

Because the length of the hypotenuse and side adjacent to are given, the cosine function can be used to find .

33.

SOLUTION:

Because the length of the hypotenuse and side opposite are given, the sine function can be used to find .

34.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

35.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

36.

SOLUTION:

Because the length of the hypotenuse and side adjacent to are given, the cosine function can be used to find .

37.

SOLUTION:

Because the length of the hypotenuse and side opposite are given, the sine function can be used to find .

38.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

39.PARASAILING Kayladecidedtotryparasailing.Shewasstrappedintoaparachutetowedbyaboat.An800-foot line connected her parachute to the boat, which was at a 32 angle of depression below her. How high above the water was Kayla?

SOLUTION:The angle of elevation from the boat to the parachute is equivalent to the angle of depression from the parachute tothe boat because the two angles are alternate interior angles, as shown below.

Because an acute angle and the hypotenuse are given, the sine function can be used to find x.

Therefore, Kayla was about 424 feet above the water.

40.OBSERVATION WHEEL TheLondonEyeisa135-meter-tall observation wheel. If a passenger at the top of the wheel sights the London Aquarium at a 58 angle of depression, what is the distance between the aquarium andthe London Eye?

SOLUTION:The angle of depression from the top of the wheel to the aquarium is 58, which means that the angle of elevation from the aquarium to the top of the wheel is also 58 because they are alternate interior angles. Draw a diagram of a right triangle and label one acute angle 58 and the opposite side 135 m.

Because an acute angle and the side length opposite the angle are given, the tangent function can be used to find x.

Therefore, the distance between the aquarium and the London Eye is about 84 meters.

41.ROLLER COASTER Onarollercoaster,375feetoftrackascendata55 angle of elevation to the top before the first and highest drop. a. Draw a diagram to represent the situation. b. Determine the height of the roller coaster.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 55 and the hypotenuse 375 feet.

b. Because an acute angle and the length of the hypotenuse are given, you can use the sine function to find the length of the opposite side.

Therefore, the height of the roller coaster is about 307 feet.

42.SKI LIFTAcompanyisinstallinganewskiliftona225-meter-high mountain that will ascend at a 48 angle of elevation. a. Draw a diagram to represent the situation. b. Determine the length of cable the lift requires to extend from the base to the peak of the mountain.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 48 and the opposite side 225 meters.

b. Because an acute angle and the length of the side opposite the angle are given, you can use the sine function to find the length of the hypotenuse.

So, the company will need about 303 meters of cable.

43.BASKETBALLBothDerekandSamare5feet10inchestall.Dereklooksata10-foot basketball goal with an angleofelevationof29,andSamlooksatthegoalwithanangleofelevationof43.IfSamisdirectlyinfrontofDerek, how far apart are the boys standing?

SOLUTION:Draw a diagram to model the situation. The vertical distance from the boys' heads to the rim is 10(12) [5(12) + 10] or 50 inches. Label the horizontal distance between Sam and Derek as x and the horizontal distance between Sam and the goal as y .

From the smaller right triangle, you can use the tangent function to find y .

From the larger right triangle, you can use the tangent function to find x.

Therefore, Derek and Sam are standing about 36.6 inches or 3.1 feet apart.

44.PARISAtouristonthefirstobservationleveloftheEiffelTowersightstheMuseDOrsay at a 1.4 angle of depression.Atouristonthethirdobservationlevel,located219metersdirectlyabovethefirst,sightstheMuseDOrsay at a 6.8 angle of depression. a. Draw a diagram to represent the situation. b. DeterminethedistancebetweentheEiffelTowerandtheMuseDOrsay.

SOLUTION:

a.

b. Because the angles of depression from the 1st

and 3rdlevelstotheMuseDOrsay are 1.4 and 6.8,

respectively,theanglesofelevationfromtheMuseDOrsaytothe1st and 3rd levels are also 1.4 and 6.8, respectively.

Use the tangent function to write an equation for the smaller right triangle in terms of y .

Next, use the tangent function to write an equation for the larger right triangle in terms of y .

Set the equations that you found for each triangle equal to one another and solve for x.

Therefore,thedistancebetweentheEiffelTowerandtheMuseDOrsay is about 2310 meters.

45.LIGHTHOUSETwoshipsarespottedfromthetopofa156-foot lighthouse. The first ship is at a 27 angle of depression, and the second ship is directly behind the first at a 7 angle of depression. a. Draw a diagram to represent the situation. b. Determine the distance between the two ships.

SOLUTION:a.

b. From the smaller right triangle, you can use the tangent function to find y .

From the larger right triangle, you can use the tangent fuction to find x.

Therefore, the distance between the two ships is about 964 feet.

46.MOUNT RUSHMOREThefacesofthepresidentsatMountRushmoreare60feettall.Avisitorseesthetopof George Washingtons head at a 48 angle of elevation and his chin at a 44.76 angle of elevation. Find the height of Mount Rushmore.

SOLUTION:From the smaller triangle, you can use the tangent function to find y .

From the larger triangle, you can use the tangent function to find y , too.

Next, set the two equations equal to one another to solve for x.

Therefore, Mount Rushmore is about 500 feet tall.

Solve each triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

47.

SOLUTION:Use trigonometric functions to find b and c.

Because the measures of two angles are given, B can be found by subtracting A from

Therefore, b 16.5,c 17.5.

48.

SOLUTION:Use trigonometric functions to find y and z .

Because the measures of two angles are given, X can be found by subtracting Z from

Therefore, y 37.1,z 32.5.

49.

SOLUTION:Use the Pythagorean Theorem to find r.

Use the tangent function to find P.

Because the measures of two angles are now known, you can find Q by subtracting P from

Therefore, P 43, Q 47, and r 34.0.

50.

SOLUTION:Use the Pythagorean Theorem to find d.

Use the cosine function to find D.

Because the measures of two angles are now known, you can find E by subtracting D from

Therefore, D 77, E 13, and d 29.2.

51.

SOLUTION:Use trigonometric functions to find j and k .

Because the measures of two angles are given, K can be found by subtracting J from

Therefore, j 18.0,k 6.2.

52.

SOLUTION:Use the Pythagorean Theorem to find y .

Use the sine function to find W.

Because the measures of two angles are now known, you can find Y by subtracting W from

Therefore, and y 3.9.

53.

SOLUTION:Use trigonometric functions to find f and h.

Because the measures of two angles are given, H can be found by subtracting F from

Therefore, f 19.6,h 17.1.

54.

SOLUTION:Use the Pythagorean Theorem to find t.

Use the tangent function to find R.

Because the measures of two angles are now known, you can find S by subtracting R from

Therefore, and t 8.1.

55.BASEBALL Michaels seat at a game is 65 feet behind home plate. His line of vision is 10 feet above the field. a. Draw a diagram to represent the situation. b. What is the angle of depression to home plate?

SOLUTION:a.

b. Michaels angle of depression to home plate is equivalent to the angle of elevation from home plate to where he is sitting.

Use the tangent function to find .

Therefore, the angle of depression to home plate is about

56.HIKING Jessica is standing 2 miles from the center of the base of Pikes Peak, and looking at the summit of the mountain, which is 1.4 miles from the base. a. Draw a diagram to represent the situation. b. With what angle of elevation is Jessica looking at the summit of the mountain?

SOLUTION:a.

b. Use the tangent function to find the angle of elevation.

Therefore, the angle of elevation is about

Find the exact value of each expression without using a calculator.57.sin 60

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side opposite the 60 angle is x and the length of the hypotenuse is 2x.

58.cot30

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side adjacent to the 30 angle is x and the length of the opposite side is x.

59.sec30

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the hypotenuse is 2x and the length of the side adjacent to the 30 angle is x.

60.cos45

SOLUTION:Draw a diagram of a 45-45-90 triangle.

The side length is x and the hypotenuse is x.

61.tan60

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side opposite the 60 is x and the length of the adjacent side is x.

62.csc45

SOLUTION:Draw a diagram of a 45-45-90 triangle.

The length of the hypotenuse is x and the side length is x.

Without using a calculator, find the measure of the acute angle that satisfies the given equation.63.tan = 1

SOLUTION:

Because tan = 1 and tan = , it follows that =1.Inthe triangle below, the side length

opposite an acute angle is 1 and the adjacent side length is also 1. So, =1.

Therefore, = 45.

64.cos =

SOLUTION:

Because cos = and cos = , it follows that = . In the triangle below, the side

length adjacent to the 30 angle is and the hypotenuse is 2. So, = .

Therefore, = 30.

65.cot =

SOLUTION:

Because cot = and cot = , it follows that = . In the triangle below, the side

length that is adjacent to the 60 angle is 1 and the length of the opposite side is . So, = or .

Therefore, = 60.

66.sin =

SOLUTION:

Because sin = and sin = , it follows that = . In the triangle below, the side

length opposite an acute angle is 1 and the hypotenuse is .So, = or .

Therefore, =45.

67.csc = 2

SOLUTION:

Because csc = 2 and csc = , it follows that =2or . In the triangle below, the

hypotenuse is 2 and the side length that is opposite the 30 angle is 1. So, = or2.

Therefore, =30.

68.sec = 2

SOLUTION:

Because sec = 2 and sec = , it follows that =2or . In the triangle below, the

hypotenuse is 2 and the side length that is adjacent to the 60 angle is 1. So, = or2.

Therefore, = 60.

Without using a calculator, determine the value of x.

69.

SOLUTION:Because an acute angle, the hypotenuse, and the length of the side adjacent to the angle are given, the cosine

function can be used to find x. Using the properties of triangles, you can find that cos 30 = .

70.

SOLUTION:

Because the triangle is a triangle, the legs are the same length.

71.SCUBA DIVINGAscubadiverlocated20feetbelowthesurfaceofthewaterspotsashipwreckata70 angle of depression. After descending to a point 45 feet above the ocean floor, the diver sees the shipwreck at a 57 angle of depression. Draw a diagram to represent the situation, and determine the depth of the shipwreck.

SOLUTION:First, draw a diagram to represent the situation.

Label the horizontal distance from the shipwreck to the point on the ocean floor below the diver as x. Label the vertical distance from the point 20 feet below sea level to the point 45 feet below sea level as y . Find the complementary angle for each angle of depression. Use the tangent function to write an equation for the smaller right triangle in terms of x.

Use the tangent function to write an equation for the larger right triangle in terms of x.

Set the two equations equal to one another to solve for y .

Therefore, the depth of the shipwreck is 20 + 35 + 45 or about 100 feet.

Find the value of cos if is the measure of the smallest angle in each type of right triangle.72.3-4-5

SOLUTION:Draw a diagram of a 3-4-5 triangle. The smallest angle will be the angle opposite the side with a length of 3.

73.5-12-13

SOLUTION:Draw a diagram of a 5-12-13 triangle. The smallest angle will be the angle opposite the side with a length of 5.

74.SOLAR POWERFindthetotalareaofthepanelshownbelow.

SOLUTION:

The area of the panel is given by , where the width is 10 feet and the length is unknown. Notice that the length of the panel is also the hypotenuse of a right triangle with an acute angle of 55 and an opposite side length of 3.5 feet. The sine function can be used to find the hypotenuse of the triangle.

So, the length of the panel is 4.27 feet. Find the area.

Therefore, the area of the panel is 42.7 square feet.

Without using a calculator, insert the appropriate symbol >, cot 60.

76.tan60 cot30

SOLUTION:

Use a graphing calculator to find tan 60 and cot 30. To find cot 30, find .

tan 60 1.732 cot 30 1.732 Therefore,tan60=cot30.

77.cos30 csc45

SOLUTION:

Use a graphing calculator to find cos 30 and csc 45. To find csc 45, find .

cos 30 0.866 csc 45 1.414 Therefore,cos30 csc60.

80.tan45 sec30

SOLUTION:tan45sec30

Use a graphing calculator to find tan 45 and sec 30. To find sec 30, find .

tan 45 = 1 sec 30 1.155 Therefore,tan45

Find the exact values of the six trigonometric functions of .

1.

SOLUTION:

The length of the side opposite is 8 , the length of the side adjacent to is 14, and the length of the hypotenuseis 18.

2.

SOLUTION:

The length of the side opposite is 2 , the length of the side adjacent to is 13, and the length of the hypotenuse is 15.

3.

SOLUTION:

The length of the side opposite is 9, the length of the side adjacent to is 4, and the length of the hypotenuse is

.

4.

SOLUTION:

The length of the side opposite is 12, the length of the side adjacent to is 35, and the length of the hypotenuse is 37.

5.

SOLUTION:

The length of the side opposite is , the length of the side adjacent to is 26, and the length of the hypotenuse is 29.

6.

SOLUTION:

The length of the side opposite is 30, the length of the side adjacent to is 5 , and the length of the hypotenuse is 35.

7.

SOLUTION:

The length of the side opposite is 6 and the length of the hypotenuse is 10. By the Pythagorean Theorem, the

length of the side adjacent to is = or8.

8.

SOLUTION:

The length of the side opposite is 8 and the length of the side adjacent to is 32. By the Pythagorean Theorem, the length of the hypotenuse is

= or8 .

Use the given trigonometric function value of the acute angle to find the exact values of the five remaining trigonometric function values of .

9.sin =

SOLUTION:

Draw a right triangle and label one acute angle . Because sin = = , label the opposite side 4 and the

hypotenuse 5.

By the Pythagorean Theorem, the length of the side adjacent to is or3.

10.cos =

SOLUTION:

Draw a right triangle and label one acute angle . Because cos = = , label the adjacent side 6 and the

hypotenuse 7.

By the Pythagorean Theorem, the length of the side opposite is or .

11.tan =3

SOLUTION:

Draw a right triangle and label one acute angle . Because tan = =3 or , label the side opposite 3 and

the adjacent side 1.

By the Pythagorean Theorem, the length of the hypotenuse is or .

12.sec =8

SOLUTION:

Draw a right triangle and label one acute angle . Because sec = =8or , label the hypotenuse 8 and

the adjacent side 1.

By the Pythagorean Theorem, the length of the side adjacent to is = or3 .

13.cos =

SOLUTION:

Draw a right triangle and label one acute angle . Because cos = = , label the adjacent side 5 and the

hypotenuse 9.

By the Pythagorean Theorem, the length of the side opposite is or2 .

14.tan =

SOLUTION:

Draw a right triangle and label one acute angle . Because tan = = , label the side opposite 5 and

adjacent side 4.

By the Pythagorean Theorem, the length of the hypotenuse is or .

15.cot =5

SOLUTION:

Draw a right triangle and label one acute angle . Because cot = =5 or ,labelthesideadjacentto as

5 and the opposite side 1.

By the Pythagorean Theorem, the length of the hypotenuse is or .

16.csc =6

SOLUTION:

Draw a right triangle and label one acute angle . Because csc = =6 or , label the hypotenuse 6 and the

side opposite as 1.

By the Pythagorean Theorem, the length of the adjacent side is or .

17.sec =

SOLUTION:

Draw a right triangle and label one acute angle . Because sec = = , label the hypotenuse 9 and the side

opposite as 2.

By the Pythagorean Theorem, the length of the opposite side is or .

18.sin =

SOLUTION:

Draw a right triangle and label one acute angle . Because sin = = , label the side opposite as 8 and

the hypotenuse 13.

By the Pythagorean Theorem, the length of the adjacent side is or .

Find the value of x. Round to the nearest tenth.

19.

SOLUTION:An acute angle measure and the length of the hypotenuse are given, so the sine function can be used to find the

length of the side opposite .

20.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the tangent function can be used to find the length of the side adjacent to .

21.

SOLUTION:An acute angle measure and the length of the hypotenuse are given, so the cosine function can be used to find the

length of the side adjacent to .

22.

SOLUTION:

An acute angle measure and the length of the side adjacent to are given, so the cosine function can be used to find the length of the hypotenuse.

23.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the sine function can be used to find thelength of the hypotenuse.

24.

SOLUTION:

An acute angle measure and the length of the side adjacent to are given, so the tangent function can be used to find the length of the side opposite .

25.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the tangent function can be used to find the length of the side adjacent to .

26.

SOLUTION:

An acute angle measure and the length of the side opposite are given, so the sine function can be used to find thelength of the hypotenuse.

27.MOUNTAIN CLIMBINGAteamofclimbersmustdeterminethewidthofaravineinordertosetupequipment to cross it. If the climbers walk 25 feet along the ravine from their crossing point, and sight the crossing point on the far side of the ravine to be at a 35 angle, how wide is the ravine?

SOLUTION:

An acute angle measure and the adjacent side length are given, so the tangent function can be used to find the length of the opposite side.

Therefore, the ravine is about 17.5 feet wide.

28.SNOWBOARDINGBradbuiltasnowboardingrampwithaheightof3.5feetandan18 incline. a. Draw a diagram to represent the situation. b. Determine the length of the ramp.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 18 and the opposite side 3.5 feet.

b. Because an acute angle measure and opposite side length are given, the sine function can be used to find the length of the hypotenuse.

Therefore, the length of the ramp is about 11.3 feet.

29.DETOURTrafficisdetouredfromElwoodAve.,left0.8mileonMapleSt.,andthenrightonOakSt.,whichintersectsElwoodAve.ata32angle. a. Draw a diagram to represent the situation. b. Determine the length of Elwood Ave. that is detoured.

SOLUTION:a. Draw a right triangle with acute angle 32 and opposite side length 0.8 mile. Label the side opposite the 32 angleMaple St., the hypotenuse Oak St., and the adjacent side Elwood Ave.

b. Because an acute angle and the length of opposite side are given, the tangent function can be used to find the adjacent side length.

Therefore, the length of Elwood Ave. that is detoured is about 1.3 miles.

30.PARACHUTINGAparatrooperencountersstrongerwindsthananticipatedwhileparachutingfrom1350feet,causing him to drift at an 8 angle. How far from the drop zone will the paratrooper land?

SOLUTION:Because an acute angle and the length of the side that is adjacent to the angle are given, the tangent function can be used to find the length of the opposite side.

So, the paratrooper will land about 190 feet away from the drop zone.

Find the measure of angle . Round to the nearest degree, if necessary.

31.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

32.

SOLUTION:

Because the length of the hypotenuse and side adjacent to are given, the cosine function can be used to find .

33.

SOLUTION:

Because the length of the hypotenuse and side opposite are given, the sine function can be used to find .

34.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

35.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

36.

SOLUTION:

Because the length of the hypotenuse and side adjacent to are given, the cosine function can be used to find .

37.

SOLUTION:

Because the length of the hypotenuse and side opposite are given, the sine function can be used to find .

38.

SOLUTION:

Because the lengths of the sides opposite and adjacent to are given, the tangent function can be used to find .

39.PARASAILING Kayladecidedtotryparasailing.Shewasstrappedintoaparachutetowedbyaboat.An800-foot line connected her parachute to the boat, which was at a 32 angle of depression below her. How high above the water was Kayla?

SOLUTION:The angle of elevation from the boat to the parachute is equivalent to the angle of depression from the parachute tothe boat because the two angles are alternate interior angles, as shown below.

Because an acute angle and the hypotenuse are given, the sine function can be used to find x.

Therefore, Kayla was about 424 feet above the water.

40.OBSERVATION WHEEL TheLondonEyeisa135-meter-tall observation wheel. If a passenger at the top of the wheel sights the London Aquarium at a 58 angle of depression, what is the distance between the aquarium andthe London Eye?

SOLUTION:The angle of depression from the top of the wheel to the aquarium is 58, which means that the angle of elevation from the aquarium to the top of the wheel is also 58 because they are alternate interior angles. Draw a diagram of a right triangle and label one acute angle 58 and the opposite side 135 m.

Because an acute angle and the side length opposite the angle are given, the tangent function can be used to find x.

Therefore, the distance between the aquarium and the London Eye is about 84 meters.

41.ROLLER COASTER Onarollercoaster,375feetoftrackascendata55 angle of elevation to the top before the first and highest drop. a. Draw a diagram to represent the situation. b. Determine the height of the roller coaster.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 55 and the hypotenuse 375 feet.

b. Because an acute angle and the length of the hypotenuse are given, you can use the sine function to find the length of the opposite side.

Therefore, the height of the roller coaster is about 307 feet.

42.SKI LIFTAcompanyisinstallinganewskiliftona225-meter-high mountain that will ascend at a 48 angle of elevation. a. Draw a diagram to represent the situation. b. Determine the length of cable the lift requires to extend from the base to the peak of the mountain.

SOLUTION:a. Draw a diagram of a right triangle and label one acute angle 48 and the opposite side 225 meters.

b. Because an acute angle and the length of the side opposite the angle are given, you can use the sine function to find the length of the hypotenuse.

So, the company will need about 303 meters of cable.

43.BASKETBALLBothDerekandSamare5feet10inchestall.Dereklooksata10-foot basketball goal with an angleofelevationof29,andSamlooksatthegoalwithanangleofelevationof43.IfSamisdirectlyinfrontofDerek, how far apart are the boys standing?

SOLUTION:Draw a diagram to model the situation. The vertical distance from the boys' heads to the rim is 10(12) [5(12) + 10] or 50 inches. Label the horizontal distance between Sam and Derek as x and the horizontal distance between Sam and the goal as y .

From the smaller right triangle, you can use the tangent function to find y .

From the larger right triangle, you can use the tangent function to find x.

Therefore, Derek and Sam are standing about 36.6 inches or 3.1 feet apart.

44.PARISAtouristonthefirstobservationleveloftheEiffelTowersightstheMuseDOrsay at a 1.4 angle of depression.Atouristonthethirdobservationlevel,located219metersdirectlyabovethefirst,sightstheMuseDOrsay at a 6.8 angle of depression. a. Draw a diagram to represent the situation. b. DeterminethedistancebetweentheEiffelTowerandtheMuseDOrsay.

SOLUTION:

a.

b. Because the angles of depression from the 1st

and 3rdlevelstotheMuseDOrsay are 1.4 and 6.8,

respectively,theanglesofelevationfromtheMuseDOrsaytothe1st and 3rd levels are also 1.4 and 6.8, respectively.

Use the tangent function to write an equation for the smaller right triangle in terms of y .

Next, use the tangent function to write an equation for the larger right triangle in terms of y .

Set the equations that you found for each triangle equal to one another and solve for x.

Therefore,thedistancebetweentheEiffelTowerandtheMuseDOrsay is about 2310 meters.

45.LIGHTHOUSETwoshipsarespottedfromthetopofa156-foot lighthouse. The first ship is at a 27 angle of depression, and the second ship is directly behind the first at a 7 angle of depression. a. Draw a diagram to represent the situation. b. Determine the distance between the two ships.

SOLUTION:a.

b. From the smaller right triangle, you can use the tangent function to find y .

From the larger right triangle, you can use the tangent fuction to find x.

Therefore, the distance between the two ships is about 964 feet.

46.MOUNT RUSHMOREThefacesofthepresidentsatMountRushmoreare60feettall.Avisitorseesthetopof George Washingtons head at a 48 angle of elevation and his chin at a 44.76 angle of elevation. Find the height of Mount Rushmore.

SOLUTION:From the smaller triangle, you can use the tangent function to find y .

From the larger triangle, you can use the tangent function to find y , too.

Next, set the two equations equal to one another to solve for x.

Therefore, Mount Rushmore is about 500 feet tall.

Solve each triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

47.

SOLUTION:Use trigonometric functions to find b and c.

Because the measures of two angles are given, B can be found by subtracting A from

Therefore, b 16.5,c 17.5.

48.

SOLUTION:Use trigonometric functions to find y and z .

Because the measures of two angles are given, X can be found by subtracting Z from

Therefore, y 37.1,z 32.5.

49.

SOLUTION:Use the Pythagorean Theorem to find r.

Use the tangent function to find P.

Because the measures of two angles are now known, you can find Q by subtracting P from

Therefore, P 43, Q 47, and r 34.0.

50.

SOLUTION:Use the Pythagorean Theorem to find d.

Use the cosine function to find D.

Because the measures of two angles are now known, you can find E by subtracting D from

Therefore, D 77, E 13, and d 29.2.

51.

SOLUTION:Use trigonometric functions to find j and k .

Because the measures of two angles are given, K can be found by subtracting J from

Therefore, j 18.0,k 6.2.

52.

SOLUTION:Use the Pythagorean Theorem to find y .

Use the sine function to find W.

Because the measures of two angles are now known, you can find Y by subtracting W from

Therefore, and y 3.9.

53.

SOLUTION:Use trigonometric functions to find f and h.

Because the measures of two angles are given, H can be found by subtracting F from

Therefore, f 19.6,h 17.1.

54.

SOLUTION:Use the Pythagorean Theorem to find t.

Use the tangent function to find R.

Because the measures of two angles are now known, you can find S by subtracting R from

Therefore, and t 8.1.

55.BASEBALL Michaels seat at a game is 65 feet behind home plate. His line of vision is 10 feet above the field. a. Draw a diagram to represent the situation. b. What is the angle of depression to home plate?

SOLUTION:a.

b. Michaels angle of depression to home plate is equivalent to the angle of elevation from home plate to where he is sitting.

Use the tangent function to find .

Therefore, the angle of depression to home plate is about

56.HIKING Jessica is standing 2 miles from the center of the base of Pikes Peak, and looking at the summit of the mountain, which is 1.4 miles from the base. a. Draw a diagram to represent the situation. b. With what angle of elevation is Jessica looking at the summit of the mountain?

SOLUTION:a.

b. Use the tangent function to find the angle of elevation.

Therefore, the angle of elevation is about

Find the exact value of each expression without using a calculator.57.sin 60

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side opposite the 60 angle is x and the length of the hypotenuse is 2x.

58.cot30

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side adjacent to the 30 angle is x and the length of the opposite side is x.

59.sec30

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the hypotenuse is 2x and the length of the side adjacent to the 30 angle is x.

60.cos45

SOLUTION:Draw a diagram of a 45-45-90 triangle.

The side length is x and the hypotenuse is x.

61.tan60

SOLUTION:Draw a diagram of a 30-60-90 triangle.

The length of the side opposite the 60 is x and the length of the adjacent side is x.

62.csc45

SOLUTION:Draw a diagram of a 45-45-90 triangle.

The length of the hypotenuse is x and the side length is x.

Without using a calculator, find the measure of the acute angle that satisfies the given equation.63.tan = 1

SOLUTION:

Because tan = 1 and tan = , it follows that =1.Inthe triangle below, the side length

opposite an acute angle is 1 and the adjacent side length is also 1. So, =1.

Therefore, = 45.

64.cos =

SOLUTION:

Because cos = and cos = , it follows that = . In the triangle below, the side

length adjacent to the 30 angle is and the hypotenuse is 2. So, = .

Therefore, = 30.

65.cot =

SOLUTION:

Because cot = and cot = , it follows that = . In the triangle below, the side

length that is adjacent to the 60 angle is 1 and the length of the opposite side is . So, = or .

Therefore, = 60.

66.sin =

SOLUTION:

Because sin = and sin = , it follows that = . In the triangle below, the side

length opposite an acute angle is 1 and the hypotenuse is .So, = or .

Therefore, =45.

67.csc = 2

SOLUTION:

Because csc = 2 and csc = , it follows that =2or . In the triangle below, the

hypotenuse is 2 and the side length that is opposite the 30 angle is 1. So, = or2.

Therefore, =30.

68.sec = 2

SOLUTION:

Because sec = 2 and sec = , it follows that =2or . In the triangle below, the

hypotenuse is 2 and the side length that is adjacent to the 60 angle is 1. So, = or2.

Therefore, = 60.

Without using a calculator, determine the value of x.

69.

SOLUTION:Because an acute angle, the hypotenuse, and the length of the side adjacent to the angle are given, the cosine

function can be used to find x. Using the properties of triangles, you can find that cos 30 = .

70.

SOLUTION:

Because the triangle is a triangle, the legs are the same length.

71.SCUBA DIVINGAscubadiverlocated20feetbelowthesurfaceofthewaterspotsashipwreckata70 angle of depression. After descending to a point 45 feet above the ocean floor, the diver sees the shipwreck at a 57 angle of depression. Draw a diagram to represent the situation, and determine the depth of the shipwreck.

SOLUTION:First, draw a diagram to represent the situation.

Label the horizontal distance from the shipwreck to the point on the ocean floor below the diver as x. Label the vertical distance from the point 20 feet below sea level to the point 45 feet below sea level as y . Find the complementary angle for each angle of depression. Use the tangent function to write an equation for the smaller right triangle in terms of x.

Use the tangent function to write an equation for the larger right triangle in terms of x.

Set the two equations equal to one another to solve for y .

Therefore, the depth of the shipwreck is 20 + 35 + 45 or about 100 feet.

Find the value of cos if is the measure of the smallest angle in each type of right triangle.72.3-4-5

SOLUTION:Draw a diagram of a 3-4-5 triangle. The smallest angle will be the angle opposite the side with a length of 3.

73.5-12-13

SOLUTION:Draw a diagram of a 5-12-13 triangle. The smallest angle will be the angle opposite the side with a length of 5.

74.SOLAR POWERFindthetotalareaofthepanelshownbelow.

SOLUTION:

The area of the panel is given by , where the width is 10 feet and the length is unknown. Notice that the length of the panel is also the hypotenuse of a right triangle with an acute angle of 55 and an opposite side length of 3.5 feet. The sine function can be used to find the hypotenuse of the triangle.

So, the length of the panel is 4.27 feet. Find the area.

Therefore, the area of the panel is 42.7 square feet.

Without using a calculator, insert the appropriate symbol >, cot 60.

76.tan60 cot30

SOLUTION:

Use a graphing calculator to find tan 60 and cot 30. To find cot 30, find .

tan 60 1.732 cot 30 1.732 Therefore,tan60=cot30.

77.cos30 csc45

SOLUTION:

Use a graphing calculator to find cos 30 and csc 45. To find csc 45, find .

cos 30 0.866 csc 45 1.414 Therefore,cos30 csc60.

80.tan45 sec30

SOLUTION:tan45sec30

Use a graphing calculator to find tan 45 and sec 30. To find sec 30, find .

tan 45 = 1 sec 30 1.155 Therefore,tan45

Find the exact values of the six trigonometric functions of .

1.

SOLUTION:

The length of the side opposite is 8 , the length of the side adjacent to is 14, and the length of the hypotenuseis 18.

2.

SOLUTION:

The length of the side opposite is 2 , the length of the side adjacent to is 13, and the length of the hypotenuse is 15.

3.

SOLU