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. x 2 + y 2 √ r = (–4) 2 + 3 2 √ = = 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 cos θ = x r = 0 1 = 0 undefined cot θ = x y = 0 –1 tan θ = y x = –1 0 sec θ = r x = 1 0 sin θ = y r 1 = 1 – = –1 csc θ = r y = 1 1 – = –1 = 0

5. 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. x 2 + y 2 √ r = 3 2 + (–3) 2 √ = = 18√ = 3√ 2

6. GUIDED PRACTICE for Examples 1 and 2 sin θ = y r cos θ = x r tan θ = y x = 3 3 – csc θ = r y sec θ = r x cot θ = x y = 3 3 – Using x = 3, y = –3, and r = 3√ 2, you can write the following: = 3 – 3√ 23√ 2 = – 3√ 23√ 2 3 = –1 = 3√ 23√ 2 3 – = –√ 2 3√ 23√ 2 3 = = √ 2 = –1 = – 2 √ 2 = 2

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

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

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

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

11. GUIDED PRACTICE for Examples 1 and 2 4. Use the unit circle to evaluate the six trigonometric functions of θ = 180°. 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. SOLUTION

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