1. 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

2. EXAMPLE 1
Evaluate trigonometric functions given a point
Using x = –4, y = 3, and r = 5, you can write the following:
sin θ = y r = 3 5
cos θ = x r = 4 5 –
tan θ = y x = 3 4 –
csc θ = r y = 5 3
sec θ = r x = 5 4 –
cot θ = x y = 4 3 –

3. 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.

4. EXAMPLE 2 Use the unit circle sin θ = y r 1 = – csc θ = r y = 1 – = –1
cos θ = x r = 1
sec θ = r x = 1
= 0
undefined
tan θ = y x = –1
cot θ = x y = –1
undefined
= 0

5. EXAMPLE 3
Find reference angles
Find the reference angle θ' for (a) θ = 5π 3
and (b) θ = – 130°.
SOLUTION a.
The terminal side of θ lies in Quadrant IV.
So, θ' = 2π – 5π 3 π = b.
Note that θ is coterminal with 230°, whose terminal side lies in Quadrant III. So, θ' = 230° – 180° + 50°.

6. EXAMPLE 4
Use reference angles to evaluate functions
Evaluate (a) tan ( – 240°) and (b) csc
17π 6
SOLUTION 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:
tan (–240°) = – tan 60° = – √ 3

7. 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:
17π 6 5π π
csc = csc = 2
17π 6 5π

8. 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 32
sin 2θ
How far can the frogbot jump on Earth?

9. Calculate horizontal distance traveled
EXAMPLE 5
Calculate horizontal distance traveled
SOLUTION
d = v2 32
sin 2θ
Write model for horizontal distance.
d =
162 32
sin (2 45°)
Substitute 16 for v and 45° for θ.
= 8
Simplify.
The frogbot can jump a horizontal distance of 8 feet on Earth.

10. EXAMPLE 1
Evaluate inverse trigonometric functions
Evaluate the expression in both radians and degrees. a.
cos–1 3 2 √
SOLUTION a.
When 0 θ π or 0° °, the angle whose cosine is ≤ θ 3 2 √
cos–1 3 2 √ θ = π 6
cos–1 3 2 √ θ =
30°

11. EXAMPLE 1
Evaluate inverse trigonometric functions
Evaluate the expression in both radians and degrees. b.
sin–1 2
SOLUTION
sin–1 b.
There is no angle whose sine is 2. So, is undefined. 2

12. 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: π 2 √ 3
( – )
tan–1 3 √ θ = π –
( – )
tan–1 3 √ θ =
–60°

13. EXAMPLE 2
Solve a trigonometric equation
Solve the equation sin θ = – where 180° < θ < 270°. 5 8
SOLUTION
STEP 1
sine is – is sin– – 38.7°. This 5 8 –
Use a calculator to determine that in the
interval –90° θ °, the angle whose ≤
angle is in Quadrant IV, as shown.

14. EXAMPLE 2
Solve a trigonometric equation
STEP 2
Find the angle in Quadrant III (where
180° < θ < 270°) that has the same sine
value as the angle in Step 1. The angle is: θ
180° ° = 218.7°
CHECK :
Use a calculator to check the answer. 5 8
sin 218.7°
– 0.625 = – 

15. 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 =
adj
hyp 6 11
cos – 1 θ = 6 11
56.9°
The correct answer is C.
ANSWER

16. 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?

17. 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 =
opp
adj 8 20

18. EXAMPLE 4
Write and solve a trigonometric equation
STEP 3
Use: a calculator to find the measure of θ.
tan–1 θ = 8 20
21.8°
The angle of the ramp is about 22°.
ANSWER