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Find the sine, cosine, and tangent.

π. 1. 2. sin : 1 , cos : 0 , tan : undefined. ANSWER. sin : 0 , cos : 1, tan : 0. ANSWER. Find the sine, cosine, and tangent. 2. 2π. about 13.3 radians. ANSWER. Find the sine, cosine, and tangent.

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Find the sine, cosine, and tangent.

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  1. π 1. 2 sin: 1, cos: 0, tan: undefined ANSWER sin: 0, cos: 1, tan: 0 ANSWER Find the sine, cosine, and tangent. 2. 2π

  2. about 13.3 radians ANSWER Find the sine, cosine, and tangent. 3. The diameter of a wheel is 27 inches. Through how many radians does a point on the wheel move when the wheel moves 15 feet?

  3. π π 2 2 = = 1 b 1 1 3 π 4 2 4 ( , 4) a. The amplitude is a = 4 and the period is 2π. 2 3π ( 2π, 0) Intercepts: (0, 0); (π,0); (2π,0) = ( , – 4) 2 ( 2π, 4) Maximum: = Minimum: ( 2π, – 4) = EXAMPLE 1 Graph sine and cosine functions Graph (a)y = 4 sin x and(b)y = cos 4x. SOLUTION

  4. π π 2 2 = = . 4 b 3 1 1 π π π π 4 4 2 ( , –1) b. The amplitude is a = 1 and the period is 2 2 2 2 π 3π Intercepts: ( , 0) ( , 0) = ( , 0); = ( , 0) 8 8 π (0, 1); Maximums: ( , 1) 2 π Minimum: = ( , –1) 4 EXAMPLE 1 Graph sine and cosine functions Graph (a)y = 4 sin x and(b)y = cos 4x. SOLUTION

  5. π π 2 2 = = 2π 1 b The amplitude is a = 2 and the period is 1 π 3π 1 3 2 Intercepts: ( 2π, –2) ( 2π, 0) ( 2π, 0) = ( , 0); = ( , 0) 4 4 2 2 (0, 2); Maximums: ( , 2) 2π Minimum: π = ( , –2) for Example 1 YOU TRY Graph the function. 1. y = 2 cosx SOLUTION

  6. Graph the function. 2. y = 5 sinx π π 2 2 = = 2π 1 b The amplitude is a = 5 and the period is 1 3 1 π 4 4 Intercepts: (0, 0) (π, 0) = ( 2π, 0) = = ( , 0) 2π 2 2 ( 2π, 5) = Maximums: ( , 5) 3π Minimum: ( 2π, –5) = ( , –5) 2 for Example 1 YOU TRY SOLUTION

  7. Graph the function. 3. f (x) = sinπx π π 2 2 = = 2 b π The amplitude is a = 1 and the period is 1 3 1 1 4 4 Intercepts: (0, 0) (1, 0); (2, 0) = ( 2, 0) = 2 2 ( 2, 1 = Maximums: ( , 1) 3 Minimum: ( 2, –1) = ( , –1) 2 for Example 1 YOU TRY SOLUTION

  8. Graph the function. 4. g(x) = cos 4πx π π 2 2 = = b 4π The amplitude is a = 1 and the period is 1 1 1 3 1 1 3 1 1 1 2 Intercepts: ( , 0) = ( , 0); ( , 0) = ( , 0) 4 2 8 4 2 8 2 2 2 Maximums: ( , 1) ( 0, 1); 1 Minimum: ( , –1) = ( , –1) 4 for Example 1 YOU TRY SOLUTION

  9. π x. Graphy = cos 2 π 2 π 2 = = b 2π 1 1 3 1 1 1 1 ( 1, 0) = ( , 0); 4 The amplitude is a = 1. and the period is 2 2 2 4 2 4 ( 1 , 0) ( 1, – ) 3 Intercepts: = ( , 0) 4 1 1 2 (0, ) ; Maximums: (1, ) 2 1 1 Minimum: = ( , – ) 2 2 EXAMPLE 2 Graph a cosine function SOLUTION

  10. Audio Test A sound consisting of a single frequency is called a pure tone. An audiometer produces pure tones to test a person’s auditory functions. Suppose an audiometer produces a pure tone with a frequency fof 2000 hertz (cycles per second). The maximum pressure Pproduced from the pure tone is 2 millipascals. Write and graph a sine model that gives the pressure Pas a function of the time t(in seconds). EXAMPLE 3 Model with a sine function

  11. 1 b π 4000 = b 2000 = π 2 frequency = period EXAMPLE 3 Model with a sine function SOLUTION Find the values of aand bin the model P = asinbt. The maximum pressure is 2, so a = 2. You can use the frequency fto find b. STEP 1 The pressure Pas a function of time tis given by P = 2sin4000πt.

  12. Graph the model. The amplitude is a = 2 and the period is = 3 1 1 1 2 f 4 4 1 ( , 2) ( , 0) ( , –2) (0 , 0); Intercepts: = ( , 0) ; 1 1 1 1 4000 ( , 0) 2000 2000 2000 2000 1 Maximum: = ( , 2) 1 8000 2000 Minimum: 3 = ( , –2) 8000 EXAMPLE 3 Model with a sine function STEP 2

  13. 5. y = sinπx π 2 π 2 = = b π The amplitude is a = 2. and the period is 1 1 1 1 1 1 3 1 1 , 3 , 1 2 2 4 4 4 4 4 4 2 4 4 Intercepts: ( (0 , 0); ) 2, 0 = (1, 0) ; (2, 0) Maximums: ( 2, ) ( ) = – – ( 2, ) ( ) Minimum: = for Examples 2 and 3 YOU TRY Graph the function. SOLUTION

  14. Graph the function. 6. y = cosπx π 2 π 2 = = b π The amplitude is a = 2. and the period is 1 1 1 1 1 3 3 1 1 1 1 4 3 4 2 3 2 3 3 2 3 3 , , ( ) ( Intercepts: ( ( ); ) 2, 0 ) 0 2, 0 0 = = ( 0, ); ( 2, ) Maximums: – – ( 2, ) ( ) Minimum: = 1, for Examples 2 and 3 YOU TRY SOLUTION

  15. π 2 π 2 = b 3 The amplitude is a = 2 and the period is π 1 π 3 π 1 2π 2π 2π 2π 3 4 2 6 2 4 , , , , , ) ) ); ( (0 , 0); ( 0 0 0 2 2 ( Intercepts: = 3 3 3 3 ( ) ) Maximums: ( = – ) ( – ) ( Minimum: 2 2 = for Examples 2 and 3 YOU TRY Graph the function. 7. f (x) = 2 sin 3x SOLUTION

  16. Graph the function. 8. g(x) = 3 cos 4x π 2 π 2 = = b 4 The amplitude is a = 3 and the period is 1 π π 3 π 1 π π π π 3π 4 8 4 4 2 2 2 , , , , , ( ) ); ( ) ) 3 0 0 2 0 ( ( Intercepts: = = 2 2 2 8 ) Maximums: (0 , 3); ( – ) ( – ) ( Minimum: 3 3 = for Examples 2 and 3 YOU TRY SOLUTION

  17. 9. What If ? In Example 3, how would the function change if the audiometer produced a pure tone with a frequency of 1000 hertz? 1 b π 2000 = b 1000 = π 2 period frequency = for Examples 2 and 3 YOU TRY SOLUTION Find the values of aand bin the model P = asinbt. The maximum pressure is 2, so a = 2. You can use the frequency fto find b. STEP 1 The pressure Pas a function of time tis given by P = 2sin2000πt.

  18. Graph the model. The amplitude is a = 2 and the period is = 1 1 = ( , 0) ; ( , 0) (0 , 0); Intercepts: 2000 1000 1 1 3 1 4 2 4 f 1 ( , 0) ( , 2) ( , –2) = ( , 2) Maximum: 1 1 1 4000 1000 1000 1000 3 Minimum: = ( , –2) 4000 f 2000 for Examples 2 and 3 YOU TRY STEP 2 The period would increase because the frequency is decreasedp = 2 sin 2000πt

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