EXAMPLE 1 Evaluate trigonometric functions given a point

Let (–4, 3) be a point on the terminal side of an angle θ in standard position. Evaluate the six trigonometric functions of θ.

SOLUTION

Use the Pythagorean theorem to find the value of r.

x2 + y2√r = (–4)2 + 32√= = 25√ = 5

EXAMPLE 1 Evaluate trigonometric functions given a point

Using x = –4, y = 3, and r = 5, you can write the following:

sin θ =yr =

35 cos θ =

xr =

45

–

tan θ =yx =

34

– csc θ =ry =

53

sec θ =rx =

54

– cot θ =xy =

43

–

EXAMPLE 2 Use the unit circle

Use the unit circle to evaluate the six trigonometric functions of = 270°.θ

SOLUTION

Draw the unit circle, then draw the angle θ = 270° in standard position. The terminal side of θ intersects the unit circle at (0, –1), so use x = 0 and y = –1 to evaluate the trigonometric functions.

EXAMPLE 2 Use the unit circle

cos θ =xr =

01 = 0 undefined

undefined cot θ =xy =

0 –1tan θ =

yx =

–10

sec θ =rx =

10

sin θ =yr

1= 1

–= –1 csc θ =

ry =

11– = –1

= 0

EXAMPLE 3 Find reference angles

Find the reference angle θ' for (a) θ = 5π3

and (b) θ = – 130°.

SOLUTION

a. The terminal side of θ lies in Quadrant IV.

So, θ' = 2π – . 5π3

π3

=

b. Note that θ is coterminal with 230°, whose terminal side lies in Quadrant III. So, θ' = 230° – 180° + 50°.

EXAMPLE 4 Use reference angles to evaluate functions

Evaluate (a) tan ( – 240°) and (b) csc .17π6

SOLUTION

tan (–240°) = – tan 60° = – √ 3

a. The angle – 240° is coterminal with 120°. The reference angle is θ' = 180° – 120° = 60°. The tangent function is negative in Quadrant II, so you can write:

EXAMPLE 4 Use reference angles to evaluate functions

b. The angle is coterminal

with . The reference

angle is θ' = π – = .

The cosecant function is positive in Quadrant II, so you can write:

17π65π

6 5π6

π6

csc = csc = 217π6

5π6

EXAMPLE 5 Calculate horizontal distance traveled

Robotics

The “frogbot” is a robot designed for exploring rough terrain on other planets. It can jump at a 45° angle and with an initial speed of 16 feet per second. On Earth, the horizontal distance d (in feet) traveled by a projectile launched at an angle θ and with an initial speed v (in feet per second) is given by:

d = v2

32sin 2θ

How far can the frogbot jump on Earth?

EXAMPLE 5 Calculate horizontal distance traveled

SOLUTION

d = v2

32sin 2θ

d = 162

32sin (2 45°)

= 8

Write model for horizontal distance.

Substitute 16 for v and 45° for θ.

Simplify.

The frogbot can jump a horizontal distance of 8 feet on Earth.

EXAMPLE 1 Evaluate inverse trigonometric functions

Evaluate the expression in both radians and degrees.

a. cos–1 32

√

SOLUTION

a. When 0 θ π or 0° 180°, the angle whose cosine is

≤ ≤ ≤ θ ≤32

√

cos–1 32

√θ =

π6

= cos–1 32

√θ = = 30°

EXAMPLE 1 Evaluate inverse trigonometric functions

Evaluate the expression in both radians and degrees.

b. sin–1 2

SOLUTION

sin–1b. There is no angle whose sine is 2. So, is undefined.

2

EXAMPLE 1 Evaluate inverse trigonometric functions

Evaluate the expression in both radians and degrees.

3 ( – )c. tan–1 √

SOLUTION

c. When – < θ < , or – 90° < θ < 90°, the

angle whose tangent is – is:

π2

π2

√ 3

( – )tan–1 3√θ =π3

–= ( – )tan–1 3√θ = –60° =

EXAMPLE 2 Solve a trigonometric equation

Solve the equation sin θ = – where 180° < θ < 270°.

58

SOLUTION

STEP 1

sine is – is sin–1 – 38.7°. This58

58

–

Use a calculator to determine that in the

interval –90° θ 90°, the angle whose≤ ≤

angle is in Quadrant IV, as shown.

EXAMPLE 2 Solve a trigonometric equation

STEP 2

Find the angle in Quadrant III (where180° < θ < 270°) that has the same sinevalue as the angle in Step 1. The angle is:

θ 180° + 38.7° = 218.7°

CHECK : Use a calculator to check the answer. 58

sin 218.7° – 0.625=–

EXAMPLE 3 Standardized Test Practice

SOLUTION

In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse, so use the inverse cosine function to solve for θ.

cos θ =adjhyp =

611

cos – 1θ = 611

56.9°

The correct answer is C.ANSWER

EXAMPLE 4 Write and solve a trigonometric equation

Monster Trucks

A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle θ of the ramp?

EXAMPLE 4 Write and solve a trigonometric equation

SOLUTION

STEP 1 Draw: a triangle that represents the ramp.

STEP 2 Write: a trigonometric equation that involves the ratio of the ramp’s height and horizontal length.

tan θ =oppadj =

820

EXAMPLE 4 Write and solve a trigonometric equation

STEP 3 Use: a calculator to find the measure of θ.

tan–1θ = 820

21.8°

The angle of the ramp is about 22°.

ANSWER