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 = 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 –

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 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

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°.

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

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

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?

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.

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° 180°, the angle whose cosine is ≤ θ 3 2 √ cos–1 3 2 √ θ = π 6 cos–1 3 2 √ θ = 30°

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

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°

EXAMPLE 2 Solve a trigonometric equation Solve the equation sin θ = – where 180° < θ < 270°. 5 8 SOLUTION STEP 1 sine is – is sin–1 – 38.7°. This 5 8 – 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 (where 180° < θ < 270°) that has the same sine value as the angle in Step 1. The angle is: θ 180° + 38.7° = 218.7° CHECK : Use a calculator to check the answer. 5 8 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 = adj hyp 6 11 cos – 1 θ = 6 11 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 = opp adj 8 20

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