lesson 13.1 , for use with pages 852-858

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