6royhhdfkwuldqjoh 5rxqgvlghohqjwkvwrwkh … · 2019. 2. 8. · 6royhhdfkwuldqjoh...

Solve each triangle. Round side lengths to thenearest tenth and angle measures to the nearestdegree.

1.

SOLUTION:

Use the Law of Cosines to find the missing sidelength.

Use the Law of Sines to find a missing anglemeasure.

Find the measure of .

ANSWER:

A ≈ 36°, C ≈ 52°, b ≈ 5.1

2.

SOLUTION:

Use the Law of Cosines to find the measure of thelargest angle, .

Use the Law of Sines to find the measure of angle,

.


ANSWER:

A ≈ 112°, B ≈ 40°, C ≈ 28°

eSolutions Manual - Powered by Cognero

8-7 The Law of Cosines

3. a = 5, b = 8, c = 12

SOLUTION:



.


ANSWER:

A ≈ 18°, B ≈ 29°, C ≈ 133°

4. B = 110°, a = 6, c = 3

SOLUTION:




ANSWER:

A ≈ 48°, C ≈ 22°, b ≈ 7.6



PRECISION Determine whether each triangleshould be solved by beginning with the Law ofSines or the Law of Cosines. Then solve thetriangle.

5.

SOLUTION:

Since two sides and an angle opposite one of them ofa triangle are given, the triangle should be solved bybeginning with the Law of Sines.


Use Law of Sines to find c.

ANSWER:

Sines; B ≈ 40°, C ≈ 33°,c ≈ 6.8

6.

SOLUTION:

Since two sides and their included angle of a triangleare given, the triangle should be solved by beginningwith the Law of Cosines.



ANSWER:

Cosines; A ≈ 48°, C ≈ 36°, b ≈ 6.7



7. In ∆RST, R = 35°, s = 16, and t = 9.

SOLUTION:




ANSWER:

Cosines; S ≈ 114°, T ≈ 31°, r ≈ 10.1

8. FOOTBALL In a football game, the quarterback is20 yards from Receiver A. He turns 40° to seeReceiver B, who is 16 yards away. How far apartare the two receivers?

SOLUTION:


The two receivers are about 12.9 yards apart.

ANSWER:

about 12.9 yd




9.

SOLUTION:




ANSWER:

A ≈ 70°, B ≈ 40°, c ≈ 3.0

10.

SOLUTION:




ANSWER:

A ≈ 48°, C ≈ 40°, b ≈ 18.8



11.

SOLUTION:



.


ANSWER:

A ≈ 31°, B ≈ 108°, C ≈ 41°

12.

SOLUTION:



.


ANSWER:

A ≈ 102°, B ≈ 44°, C ≈ 34°



13. A = 116°, b = 5, c = 3

SOLUTION:




ANSWER:

a ≈ 6.9, B ≈ 41°, C ≈ 23°

14. C = 80°, a = 9, b = 2

SOLUTION:




ANSWER:

c ≈ 8.9, A ≈ 87°, B ≈ 13°



15. f = 10, g = 11, h = 4

SOLUTION:



.


ANSWER:

F ≈ 65°, G ≈ 94°, H ≈ 21°

16. w = 20, x = 13, y = 12

SOLUTION:



.


ANSWER:

W ≈ 106°, X ≈ 39°, Y ≈ 35°



Determine whether each triangle should besolved by beginning with the Law of Sines or theLaw of Cosines. Then solve the triangle.

17.

SOLUTION:

Since two sides and an angle opposite on of them of atriangle are given, the triangle should be solved bybeginning with the Law of Sines.

Find the measure of

Use Law of Sines to find a.

ANSWER:

Sines; C ≈ 45°, A ≈ 85°, a ≈ 18.2

18.

SOLUTION:




ANSWER:

Cosines; s ≈ 28.9, R ≈ 42°, T ≈ 32°



19.

SOLUTION:

Since three sides of a triangle are given, the triangleshould be solved by beginning with the Law ofCosines. Find the measure of the largest angle, .


.


ANSWER:

Cosines; A ≈ 27°, B ≈ 115°, C ≈ 38°

20.

SOLUTION:

Since two angles and any side of a triangle are given,the triangle should be solved by beginning with theLaw of Sines. Find the measure of .

Find the length of m and p.

ANSWER:

Sines; N ≈ 53°, p ≈ 38.2, m ≈ 28.4



21. In ∆ABC, C = 84°, c = 7, and a = 2.

SOLUTION:

Since two sides and an angle opposite on of them of atriangle are given, the triangle should be solved bybeginning with the Law of Sines. Find the measure of angle, .

Find the measure of angle, .

Find the length of b.

ANSWER:

Sines; A ≈ 17°, B ≈ 79°, b ≈ 6.9

22. In ∆HJK, h = 18, j = 10, and k = 23.

SOLUTION:

Since three sides of a triangle are given, the triangleshould be solved by beginning with the Law ofCosines. Find the measure of the largest angle, .

Use the Law of Sines to find the measure of angle

.


ANSWER:

Cosines; H ≈ 48°, J ≈ 25°, K ≈ 107°



23. EXPLORATION Find the distance between theship and the shipwreck shown in the diagram. Roundto the nearest tenth.

SOLUTION:


The distance between the ship and the shipwreck is514.2 meters.

ANSWER:

514.2 m

24. GEOMETRY A parallelogram has side lengths 8centimeters and 12 centimeters. One angle betweenthem measures 42°. To the nearest tenth, what is thelength of the shorter diagonal?

SOLUTION:

Use the Law of Cosines to find the shorter diagonal.

The length of the shorter diagonal is 8.1 cm.

ANSWER:

8.1 cm

25. RACING A triangular cross-country course hasside lengths 1.8 kilometers, 2 kilometers, and 1.2kilometers. What are the angles formed betweeneach pair of sides?

SOLUTION:

Let a = 2, b = 1.8 and c = 1.2. Use the Law of Cosines to find the measure of thelargest angle.

Use the Law of Sines to find the measure of .


ANSWER:

81°, 36°, 63°



26. MODELING A triangular plot of farm landmeasures 0.9 by 0.5 by 1.25 miles.a. If the plot of land is fenced on the border, what willbe the angles at which the fences of the three sidesmeet? Round to the nearest degree.b. What is the area of the plot of land?

SOLUTION:

a. Let a = 1.25, b = 0.9 and c = 0.5. Use the Law of Cosines to find the measure of thelargest angle.



b. Substitute b = 0.9, c = 0.5 and A = 124º in the areaformula.

ANSWER:

a. 19°, 37°, 124°b. about 0.19 mi2

27. LAND Some land is in the shape of a triangle. Thedistances between each vertex of the triangle are 140yd, 210 yd and 300 yd, respectively. Use the Law ofCosines to find the area of the land to the nearestsquare yard.

SOLUTION:

Use the Law of Cosines to find the measure of thelargest angle.

Substitute b = 210, c = 140 and A = 117º in the areaformula.

The area of the land is 13,148 yd2.

ANSWER:

13,148 yd2

28. RIDES Two bumper cars at an amusement parkride collide as shown.

a. How far apart d were the two cars before theycollided?b. Before the collision, a third car was 10 feet fromcar 1 and 13 feet from car 2. Describe the anglesformed by cars 1, 2, and 3 before the collision.

SOLUTION:

a. Let a = 5.5, b = 7, c = d and C = 118




The distance between the two cars is about 10.7 ft. b. Let a = 13, b = 10 and c = 10.7 Use the Law of Cosines to find the measure of thelargest angle.



ANSWER:

a. about 10.7 ftb. 78°, 49°, 53°

29. PICNICS A triangular picnic area is 11 yards by 14yards by 10 yards.a. Sketch and label a drawing to represent the picnicarea.b. Describe how you could find the area of the picnicarea.c. What is the area? Round to the nearest tenth.

SOLUTION:

a.

b. Use the Law of Cosines to find the measure of

. Then use the formula Area = .

c. Use the Law of Cosines to find the measure of thelargest angle.

Substitute b = 10, c = 11 and A = 83º in the areaformula.

ANSWER:

a.

b. Sample answer: Use the Law of Cosines to findthe measure of . Then use the formula Area =

.

c. 54.6 yd2



30. WATERSPORTS A person on a personalwatercraft makes a trip from point A to point B topoint C traveling 28 miles per hour. She then returnsfrom point C back to her starting point traveling 35miles per hour. How many minutes did the entire triptake? Round to the nearest tenth.

SOLUTION:


The time taken for the trip from point A to point B is

per hour.

The time taken for the trip from point B to point C is

per hour.

The time taken for the trip from point C to point A is

per hour.

Time taken for the entire trip is 0.0246 hrs or 1.5minutes.

ANSWER:

1.5 min


31.

SOLUTION:



Use the Law of Sines to find c.

ANSWER:

B ≈ 39°, C ≈ 37°, c ≈ 7.7



32.

SOLUTION:



.

Since R is obtuse Find the measure of .

ANSWER:

R ≈ 107°, S ≈ 48°, q ≈ 16.0

33.

SOLUTION:



.


ANSWER:

F ≈ 42°, G ≈ 72°, H ≈ 66°



34. CHALLENGE Use the figure and the PythagoreanTheorem to derive the Law of Cosines. Use the hintsbelow.

• First, use the Pythagorean Theorem for ∆DBC.

• In ∆ADB, c2 = x2 + h2.

•

SOLUTION:

a2 = (b – x) 2 + h2 Use the PythagoreanTheorem for ∆DBC. = b2 – 2bx + x2 + h2 Expand (b – x) 2. = b2 – 2bx + c2 In ∆ADB, c2 = x2 + h2.

= b2 – 2b(c cos A) + c2 , so x = c cos

A. = b2 + c2 – 2bc cos A Commutative Property

ANSWER:

a2 = (b – x) 2 + h2 Use the PythagoreanTheorem for ∆DBC.

= b2 – 2bx + x2 + h2 Expand (b – x) 2.

= b2 – 2bx + c2 In ∆ADB, c2 = x2 + h2.

= b2 – 2b(c cos A) + c2 , so x = c cos

A.

= b2 + c2 – 2bc cos A Commutative Property

35. CONSTRUCT ARGUMENTS Three sides of atriangle measure 10.6 centimeters, 8 centimeters, and14.5 centimeters. Explain how to find the measure ofthe largest angle. Then find the measure of the angleto the nearest degree.

SOLUTION:

The longest side is 14.5 centimeters. Use the Law ofCosines to find the measure of the angle opposite thelongest side.

Then, the measure of the angle is 102.

ANSWER:

The longest side is 14.5 centimeters. Use the Law ofCosines to find the measure of the angle opposite thelongest side; 102°.



36. OPEN-ENDED Create an application probleminvolving right triangles and the Law of Cosines.Then solve your problem, drawing diagrams ifnecessary.

SOLUTION: A group of friends who are on a camping trip decideto go on a hike. According to the map shown, what isthe angle that is formed by the two trails that lead tothe camp?

Then the angle formed by the two trails measuresabout 56.

ANSWER: See students’ work.

37. WRITING IN MATH How do you know whichmethod to use when solving a triangle?

SOLUTION: To solve a right triangle, you can use the PythagoreanTheorem to find side lengths and trigonometric ratiosto find angle measures and side lengths. To solve a nonright triangle, you can use the Law ofSines or the Law of Cosines, depending on whatinformation is given. When two angles and a side aregiven or when two sides and an angle opposite one ofthe sides are given, you can use the Law of Sines.When two sides and an included angle or three sidesare given, you can use the Law of Cosines.

ANSWER: Sample answer: To solve a right triangle, you can usethe Pythagorean Theorem to find side lengths andtrigonometric ratios to find angle measures and sidelengths. To solve a nonright triangle, you can use theLaw of Sines or the Law of Cosines, depending onwhat information is given. When two angles and aside are given or when two sides and an angleopposite one of the sides are given, you can use theLaw of Sines. When two sides and an included angleor three sides are given, you can use the Law ofCosines.



38. A manufacturing company produces triangular-shaped support brackets. The side lengths of thebrackets measure 6 feet, 8 feet, and 5 feet. Whattrigonometric relationship represents the measure ofthe largest interior angle, angle A, formed by thesupport brackets?

A

B

C

D

SOLUTION:

So the correct choice is A.

ANSWER: A

39. In ABC, the given dimensions are in the diagram.

Which of the following trigonometric ratios in ABC

is equal to ?A sin BB cos BC sin CD cos C

SOLUTION:

Use the Law of Sines.

Thus, the correct choice is A.

ANSWER: A

40. Select all of the correct expressions used in findingthe missing lengths.

A



B

C

D

E

SOLUTION: Choice A is an incorrect application of the Law ofCosines.Choice B is a correct application of the Law ofCosines. From the equation in B, a ≈ 17. Using the Law of Sines, choice C is incorrectbecause the 17 is not an angle measure. Using the Law of Sines,

Choice D is correct. Using the Law of Sines,

Choice E is correct. The correct choices are B, D, and E.

ANSWER: B, D, and E

41. A garden has sides of 8 meters, 11 meters, and 15meters. What is the area of the garden to the nearestsquare meter?

SOLUTION: Use the Law of Cosines to find the measure of oneof the angles.To find the measure of the angle between the sidesthat are 8 m and 11 m,

Then the area can be found.

area =

area = area ≈ 43 The area is approximately 43 square meters.

ANSWER: 43 square meters

42. MULTI-STEP Carlotta paddled a kayak from adock and traveled 4 miles east. She then steered thekayak 80° north of east and traveled another 2 milesbefore she anchored. a. Draw a diagram of the scenario. b. To the nearest tenth of a mile, find the distancefrom Carlotta's starting point to the point where sheanchored the kayak. c. What law did you use to find the distance? Explainyour reasoning. d. Find the measure of the two other angles of thetriangle to the nearest degree.

SOLUTION: a.



b. From the diagram, find b. Use the Law ofCosines.

Carlotta anchored about 4.8 miles away. c. The Law of Cosines had to be used because twosides and the included angle are used. d. Use the Law of Sines.

Subtract (100 + 24) from 180 to find the measure ofthe third angle, 56.Or use the Law of Sines.

The other two angles are about 24° and 55° or 56°.

ANSWER: a.

b. 4.8 milesc. The Law of Cosines had to be used because twosides and the included angle are used.d. about 24° and 55° or 56°



43. MULTI-STEP In ABC, m∠A = 50°, m∠B = 35°,and a = 12. a. Find the measure of ∠C. b. Find b to the nearest tenth. c. Find c to the nearest tenth. d. Find the perimeter of ABC.

SOLUTION: a. So the measure of ∠C is 95°. b. To find b, use the Law of Sines.

c. To find c, use the Law of Sines.

d. The perimeter is 15.6 + 9 + 12 = 36.6.

ANSWER: 95°; 9.0; 15.6; 36.6

44. In an isosceles triangle, the vertex angle is 30° andthe base is 12 cm long. Find the perimeter of thetriangle to the nearest integer.

SOLUTION: The two base angles of the isosceles triangle areeach 75°. Use the Law of Sines to find the measure of each ofthe congruent sides.

So the perimeter of the triangle is about 23 cm + 23cm + 12 cm = 58 cm.

ANSWER: 58 cm



6royhhdfkwuldqjoh 5rxqgvlghohqjwkvwrwkh … · 2019. 2. 8. · 6royhhdfkwuldqjoh...

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