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 =opphyp =

1213

csc θ =hypopp =

1312

EXAMPLE 1 Evaluate trigonometric functions

tan θ =oppadj =

125

cot θ =adjopp =

512

cos θ =adjhyp =

513

sec θ =hypadj =

135

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 =oppadj =

33√

4=

33

33

4√

ANSWER

The correct answer is B.

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 =adjhyp

Write trigonometric equation.

32

√= x

8Substitute.

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° =oppadj sec 28º =

hypadj

tan 28º =a15

sec 28º =c15

= 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

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 =x2

Write 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

EXAMPLE 1 Draw angles in standard position

Draw an angle with the given measure in standard position.

SOLUTION

a. 240º

a. Because 240º is 60º more than 180º, the terminal side is 60º counterclockwise past the negative x-axis.

EXAMPLE 1 Draw angles in standard position

Draw an angle with the given measure in standard position.

SOLUTION

b. 500º

b. Because 500º is 140º more than 360º, the terminal side makes one whole revolution counterclockwise plus 140º more.

EXAMPLE 1 Draw angles in standard position

Draw an angle with the given measure in standard position.

SOLUTION

c. –50º

c. Because –50º is negative, the terminal side is 50º clockwise from the positive x-axis.

EXAMPLE 2 Find coterminal angles

Find one positive angle and one negative angle that are coterminal with (a) –45º and (b) 395º.

SOLUTION

a. –45º + 360º

–45º – 360º

There are many such angles, depending on what multiple of 360º is added or subtracted.

= 315º

= – 405º

EXAMPLE 2 Find coterminal angles

b. 395º – 360º

395º – 2(360º)

= 35º

= –325º

EXAMPLE 3 Convert between degrees and radians

a. 125º

Convert (a) 125º to radians and (b) – radians to degrees.

π12

25π36= radians

b.π12

–π radians

180ºπ12

–= radians( )( )= –15º

( π radians180º )= 125º

EXAMPLE 4 Solve a multi-step problem

A softball field forms a sector with the dimensions shown. Find the length of the outfield fence and the area of the field.

Softball

EXAMPLE 4 Solve a multi-step problem

SOLUTION

STEP 1

Convert the measure of the central angle to radians.

90º = 90º ( π radians180º ) =

π2

radians

EXAMPLE 4 Solve a multi-step problem

STEP 2

Find the arc length and the area of the sector.

πArc length: s = r = 180 = 90π ≈ 283 feetθ2

( )

Area: A = r2θ = (180)2 = 8100π ≈ 25,400 ft2

π2

( )12

12

The length of the outfield fence is about 283 feet. The area of the field is about 25,400 square feet.

ANSWER