EXAMPLE 1 Evaluate trigonometric functions given a point

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. r = √ x 2 + y 2 = √ (– 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: 3 4 y x – sin θ = cos θ = = 5 r r tan θ = y x sec θ = r x =– 3 4 =– 5 4 5 3 csc θ = r y = cot θ = x y 4 =– 3

EXAMPLE 2 Use the unit circle to evaluate the six trigonometric functions of θ = 270°. 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 = – 1 cos θ = x r 0 = 1 =0 tan θ = y x – 1 = 0 undefined csc θ = sec θ = r y r x cot θ = 1 = – 1 1 = 0 x y undefined 0 = – 1 = 0

GUIDED PRACTICE for Examples 1 and 2 Evaluate the six trigonometric functions of θ. 1. SOLUTION Use the Pythagorean Theorem to find the value of r. r = √ x 2 + y 2 = √ 32 + (– 3)2 = √ 18 = 3√ 2

GUIDED PRACTICE for Examples 1 and 2 Using x = 3, y = – 3 , and r = 3√ 2, you can write the following: 3 y x √ 2 – – – 3 = = sin θ = cos θ = = = 2 r 3√ 2 3 3 = – 1 tan θ = y x =– sec θ = r x 2 =√ 2 = 3√ 3 √ 2 2 csc θ = r 3√ 2 = – 3 = –√ 2 y cot θ = x y 3 =– = – 1 3

GUIDED PRACTICE for Examples 1 and 2 2. SOLUTION Use the Pythagorean theorem to find the value of r. r = √ (– 8)2 + (15)2 = √ 64 + 225 = √ 289 = 17

GUIDED PRACTICE for Examples 1 and 2 Using x = – 8, y = 15, and r = 17, you can write the following: sin θ = y r 15 = 17 tan θ = y x 15 – = 8 sec θ = r x 17 – = 8 cos θ = x r 8 = – 17 csc θ = r y 17 = 15 cot θ = x y =– 8 15

GUIDED PRACTICE for Examples 1 and 2 3. SOLUTION Use the Pythagorean theorem to find the value of r. r = √ x 2 + y 2 = √ (– 5)2 + (– 12)2 = √ 25 + 144 = 13

GUIDED PRACTICE for Examples 1 and 2 Using x = – 5, y = – 12, and r = 13, you can write the following: sin θ = y r 12 – = 13 tan θ = y x = 12 5 sec θ = r x 13 – = 5 cos θ = x r 5 – = 13 csc θ = r y 13 – = 12 cot θ = x y = 5 12

GUIDED PRACTICE for Examples 1 and 2 4. Use the unit circle to evaluate the six trigonometric functions of θ = 180°. SOLUTION Draw the unit circle, then draw the angle θ = 180° in standard position. The terminal side of θ intersects the unit circle at (– 1, 0), so use x = – 1 and y = 0 to evaluate the trigonometric functions.

GUIDED PRACTICE sin θ = y r 0 = 1 =0 tan θ = y x 0 = – 1 sec θ = r – 1 = – 1 x for Examples 1 and 2 cos θ = x – 1 = r 1 csc θ = r y – 1 = 0 undefined cot θ = x y – 1 = 0 undefined = – 1