lesson 13.1, for use with pages 852-858 in right triangle abc, a and b are the lengths of the legs...
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Lesson 13.1, For use with pages 852-858
In right triangle ABC, a and b are the lengths of the legsand c is the length of the hypotenuse. Find the missinglength. Give exact values.
1. a = 6, b = 8
2. c = 10, b = 7
ANSWER c = 10
ANSWER a = 51
3. If you walk 2.0 kilometers due east and than 1.5 kilometers due north, how far will you be from your starting point?
ANSWER 2.5 km
In right triangle ABC, a and b are the lengths of the legsand c is the length of the hypotenuse. Find the missinglength. Give exact values.
Lesson 13.1, For use with pages 852-858
Trigonometry and Angles 13.1
EXAMPLE 1 Evaluate trigonometric functions
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
13=169= √
From the Pythagorean theorem, the length of the
hypotenuse is 52 + 122√
sin θ =opp
hyp=
1213
csc θ =hyp
opp=
1312
EXAMPLE 1 Evaluate trigonometric functions
tan θ =opp
adj=
12
5cot θ =
adj
opp=
5
12
cos θ =adj
hyp=
5
13sec θ =
hyp
adj=
13
5
Draw: a right triangle with acute angle θ such that the leg opposite θ has length 4 and the hypotenuse has length 7. By the Pythagorean theorem, the length x of the other leg is
x 72 – 42√=
EXAMPLE 2 Standardized Test Practice
SOLUTION
STEP 1
33.= √
EXAMPLE 2 Standardized Test Practice
STEP 2 Find the value of tan θ.
tan θ =opp
adj=
33√
4=
33
33
4 √
ANSWER
The correct answer is B.
GUIDED PRACTICE for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ.
1.
SOLUTION
5= 25= √
From the Pythagorean theorem, the length of the
hypotenuse is 32 + 42√
sin θ =opp
hyp=
3
5csc θ =
hyp
opp=
5
3
GUIDED PRACTICE for Examples 1 and 2
tan θ =opp
adj=
3
4cot θ =
adj
opp=
4
3
cos θ =adj
hyp=
4
5sec θ =
hyp
adj=
5
4
From the Pythagorean theorem, the length of the
adjacent is 172 – 152√
GUIDED PRACTICE for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
8.= 64= √
sin θ =opp
hyp=
15
17csc θ =
hyp
opp=
17
15
2.
GUIDED PRACTICE for Examples 1 and 2
tan θ =opp
adj=
15
8cot θ =
adj
opp=
8
15
cos θ =adj
hyp=
8
17sec θ =
hyp
adj=
17
8
GUIDED PRACTICE for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
5= 25= √
sin θ =opp
hypcsc θ =
hyp
opp
3.
From the Pythagorean theorem, the length of the
adjacent is (5 22 – 52√ √
=5
5 2√ 5=
5 2√
GUIDED PRACTICE for Examples 1 and 2
tan θ =opp
adj=
5
5cot θ =
adj
opp=
5
5
cos θ =adj
hypsec θ =
hyp
adj
5=
5 2√ 5=
5 2√
= 1 = 1
EXAMPLE 3 Find an unknown side length of a right triangle
SOLUTION
Write an equation using a trigonometric function that involves the ratio of x and 8. Solve the equation for x.
Find the value of x for the right triangle shown.
cos 30º =adj
hyp Write trigonometric equation.
3
2
√=
x8 Substitute.
EXAMPLE 3 Find an unknown side length of a right triangle
34 √ = x Multiply each side by 8.
The length of the side is x = 34 √ 6.93.
ANSWER
EXAMPLE 4 Use a calculator to solve a right triangle
SOLUTION
Write trigonometric equation.
Substitute.
Solve ABC.
A and B are complementary angles,
so B = 90º – 28º
tan 28° =opp
adjsec 28º =
hyp
adj
tan 28º =a
15sec 28º =
c
15
= 68º.
EXAMPLE 4 Use a calculator to solve a right triangle
Solve for the variable.
Use a calculator.
15(tan 28º) = a 151( cos 28º ) = c
7.98 a 17.0 c
So, B = 62º, a 7.98, and c 17.0
ANSWER
GUIDED PRACTICE for Examples 3 and 4
Solve ABC using the diagram at the right and the given measurements.
5. B = 45°, c = 5
SOLUTION
Substitute.
A and B are complementary angles,
so A = 90º – 45º
cos 45° =adj
hypsin 45º =
opp
hyp
cos 45º =a
5sin 45º =
5b
Write trigonometric equation.
= 45º.
GUIDED PRACTICE for Examples 3 and 4
Solve for the variable.
Use a calculator.
5(cos 45º) = a 5(sin 45º) = b
3.54 a 3.54 b
So, A = 45º, b 3.54, and a 3.54.
ANSWER
GUIDED PRACTICE for Examples 3 and 4
SOLUTION
Substitute.
A and B are complementary angles,
so B = 90º – 32º
tan 32° =opp
adjsec 32º =
hyp
adj
tan 32º =a
10sec 32º =
10c
6. A = 32°, b = 10
Write trigonometric equation.
= 58º.
GUIDED PRACTICE for Examples 3 and 4
Solve for the variable.
Use a calculator.
10(tan 32º) = a 101( cos 32º ) = c
6.25 a 11.8 c
So, B = 58º, a 6.25, and c 11.8.
ANSWER
GUIDED PRACTICE for Examples 3 and 4
SOLUTION
Substitute.
A and B are complementary angles,
so B = 90º – 71º
cos 71° =adj
hypsin 71º =
opp
hyp
cos 71º =b
20sin 71º =
a
20
7. A = 71°, c = 20
Write trigonometric equation.
= 19º.
GUIDED PRACTICE for Examples 3 and 4
Solve for the variable.
Use a calculator.
20(cos 71º) = b
6.51 b 18.9 a
So, B = 19º, b 6.51, and a 18.9.
ANSWER
20(sin 71º) = a
GUIDED PRACTICE for Examples 3 and 4
SOLUTION
Substitute.
A and B are complementary angles,so A = 90º – 60º
sec 60° =hyp
adjtan 60º =
opp
adj
sec 60º = 7c
tan 60º =b
7
Write trigonometric equation.
8. B = 60°, a = 7
= 30º.
GUIDED PRACTICE for Examples 3 and 4
Solve for the variable.
Use a calculator.
7(tan 60º) = b 71( cos 60º ) = c
14 = c 12.1 b
So, A = 30º, c = 14, and b 12.1.
ANSWER
EXAMPLE 5 Use indirect measurement
While standing at Yavapai Point near the Grand Canyon, you measure an angle of 90º between Powell Point and Widforss Point, as shown. You then walk to Powell Point and measure an angle of 76º between Yavapai Point and Widforss Point. The distance between Yavapai Point and Powell Point is about 2 miles. How wide is the Grand Canyon between Yavapai Point and Widforss Point?
Grand Canyon
EXAMPLE 5 Use indirect measurement
SOLUTION
tan 76º =x
2Write trigonometric equation.
2(tan 76º) = x Multiply each side by 2.
8.0 ≈ x Use a calculator.
The width is about 8.0 miles.
ANSWER
EXAMPLE 6 Use an angle of elevation
A parasailer is attached to a boat with a rope 300 feet long. The angle of elevation from the boat to the parasailer is 48º. Estimate the parasailer’s height above the boat.
Parasailing
EXAMPLE 6 Use an angle of elevation
SOLUTION
sin 48º =h
300Write trigonometric equation.
300(sin 48º) = h Multiply each side by 300.
STEP 1
Draw: a diagram that represents the situation.
STEP 2
Write: and solve an equation to find the height h.
223 ≈ x Use a calculator.
The height of the parasailer above the boat is about 223 feet.
ANSWER
GUIDED PRACTICE for Examples 5 and 6
9. In Example 5, find the distance between Powell Point and Widforss Point.
Grand Canyon
SOLUTION
sec 76º =2
xWrite trigonometric equation.
Multiply each side by 2.
8.27 ≈ xUse a calculator.
21
( cos 76º ) = x
The distance is about 8.27 miles.ANSWER
Substitute for sec 76° .cos 76°
1
2 sec 76º = x
GUIDED PRACTICE for Examples 5 and 6
10. What If? In Example 6, estimate the height of the parasailer above the boat if the angle of elevation is 38°.
SOLUTION
sin 38º =h
300Write trigonometric equation.
300(sin 38º) = h Multiply each side by 300.
185 ≈ h Use a calculator.
The height of the parasailer above the boat is about 185 feet.
ANSWER
EXAMPLE 5 Use indirect measurement
While standing at Yavapai Point near the Grand Canyon, you measure an angle of 90º between Powell Point and Widforss Point, as shown. You then walk to Powell Point and measure an angle of 76º between Yavapai Point and Widforss Point. The distance between Yavapai Point and Powell Point is about 2 miles. How wide is the Grand Canyon between Yavapai Point and Widforss Point?
Grand Canyon
EXAMPLE 5 Use indirect measurement
SOLUTION
tan 76º =x
2Write trigonometric equation.
2(tan 76º) = x Multiply each side by 2.
8.0 ≈ x Use a calculator.
The width is about 8.0 miles.
ANSWER
EXAMPLE 6 Use an angle of elevation
A parasailer is attached to a boat with a rope 300 feet long. The angle of elevation from the boat to the parasailer is 48º. Estimate the parasailer’s height above the boat.
Parasailing
EXAMPLE 6 Use an angle of elevation
SOLUTION
sin 48º =h
300Write trigonometric equation.
300(sin 48º) = h Multiply each side by 300.
STEP 1
Draw: a diagram that represents the situation.
STEP 2
Write: and solve an equation to find the height h.
223 ≈ x Use a calculator.
The height of the parasailer above the boat is about 223 feet.
ANSWER
GUIDED PRACTICE for Examples 5 and 6
9. In Example 5, find the distance between Powell Point and Widforss Point.
Grand Canyon
SOLUTION
sec 76º =2
xWrite trigonometric equation.
Multiply each side by 2.
8.27 ≈ xUse a calculator.
21
( cos 76º ) = x
The distance is about 8.27 miles.ANSWER
Substitute for sec 76° .cos 76°
1
2 sec 76º = x
GUIDED PRACTICE for Examples 5 and 6
10. What If? In Example 6, estimate the height of the parasailer above the boat if the angle of elevation is 38°.
SOLUTION
sin 38º =h
300Write trigonometric equation.
300(sin 38º) = h Multiply each side by 300.
185 ≈ h Use a calculator.
The height of the parasailer above the boat is about 185 feet.
ANSWER