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EXAMPLE 1

Let (–4, 3) be a point on the terminal side of an angle θ in standard position. Evaluate the six trigonometric functions of θ. r. √. 25. x 2 + y 2. ( – 4) 2 + 3 2. √. √. =. =. =. EXAMPLE 1. Evaluate trigonometric functions given a point. SOLUTION.

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EXAMPLE 1

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  1. Let (–4, 3) be a point on the terminal side of an angle θ in standard position. Evaluate the six trigonometric functions of θ. r √ 25 x2 + y2 (–4)2 + 32 √ √ = = = EXAMPLE 1 Evaluate trigonometric functions given a point SOLUTION Use the Pythagorean theorem to find the value of r. = 5

  2. 5 4 4 3 5 3 y x – sinθ cosθ = = = = 4 3 3 4 5 5 r r y r – tanθ cscθ = = = = x y r x – secθ – cotθ = = = = x y EXAMPLE 1 Evaluate trigonometric functions given a point Using x = –4, y = 3, and r =5, you can write the following:

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

  4. 1 0 r x secθ cosθ = = = = 0 1 x r –1 0 y x tanθ cotθ = = = = 0 –1 x y 1 1 – y r sinθ cscθ = = = = – 1 1 r y EXAMPLE 2 Use the unit circle = –1 = –1 undefined = 0 undefined = 0

  5. Evaluate the six trigonometric functions of . θ 1. = 3√ 2 r √ 18 x2 + y2 32 + (–3)2 √ √ = = = for Examples 1 and 2 GUIDED PRACTICE SOLUTION Use the Pythagorean Theorem to find the value of r.

  6. Using x = 3, y =–3 , and r =3√2, you can write the following: 3 3 y x 3 3 sinθ cosθ = = – – = = 3 3 r r 3√2 3√2 y r – tanθ = = – cscθ = = x y r x secθ – cotθ = = = x y 3√2 3√2 = = –√2 = √2 – 3 = = 3 2 2 √ 2 √ 2 for Examples 1 and 2 GUIDED PRACTICE = –1 = –1

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

  8. 15 8 – = = 17 17 y x sinθ cosθ = = r r 15 17 – = = y r 8 15 tanθ cscθ = = x y r x 17 8 secθ cotθ = = – – = = x y 8 15 for Examples 1 and 2 GUIDED PRACTICE Using x = –8, y = 15, and r =17, you can write the following:

  9. 3. r x2 + y2 √ = √ 25 + 144 (–5)2 + (–12)2 √ = = for Examples 1 and 2 GUIDED PRACTICE SOLUTION Use the Pythagorean theorem to find the value of r. = 13

  10. y x 5 sinθ cosθ = = – = r r 13 y r tanθ cscθ = = 12 x y = 5 r x secθ cotθ = = x y 5 = 12 13 13 12 – – – = = = 13 12 5 for Examples 1 and 2 GUIDED PRACTICE Using x = –5, y = –12, and r =13, you can write the following:

  11. for Examples 1 and 2 GUIDED PRACTICE 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.

  12. r x secθ cosθ = = x r y x 0 tanθ cotθ = = = x y –1 0 –1 –1 –1 –1 = = = = = 1 0 1 0 1 y r sinθ cscθ = = r y for Examples 1 and 2 GUIDED PRACTICE = 0 = –1 undefined = –1 undefined

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