1. 1. 2. 2π Lesson 14.1, For use with pages 908-914
Find the sine, cosine, and tangent. 1. π 2
ANSWER
sin: 1, cos: 0, tan: undefined
2. 2π
ANSWER
sin: 0, cos: 1, tan: 0

2. Lesson 14.1, For use with pages 908-914
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?
ANSWER
about 13.3 radians

3. Graphing Trigonometry Functions 14.1

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

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

6. GUIDED PRACTICE
for Example 1
Graph the function.
1. y = 2 cos x
SOLUTION
The amplitude is a = 2 and the period is 2 b π = 1
= 2π
Intercepts:
( π, 0) 1 4
= ( , 0 ) π 2 3
= ( , 0 ) 3π
( , 2)
Maximums:
(0, 2); 2π
Minimum:
( π, –2) 1 2
= ( , –2) π 4

7. GUIDED PRACTICE
for Example 1
Graph the function.
2. y = 5 sin x
SOLUTION
The amplitude is a = 5 and the period is 2 b π = 1
= 2π
Intercepts:
(0, 0)
(π, 0)
= ( π, 0) = 1 2
= ( , 0) 2π
Maximums:
( π, 5) = 1 4
( , 5) π 2
Minimum:
( π, –5) 3 4
= ( , –5) 3π 2

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

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

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

11. EXAMPLE 3
Model with a sine function
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 f of 2000 hertz (cycles per second). The maximum pressure P produced from the pure tone is 2 millipascals. Write and graph a sine model that gives the pressure P as a function of the time t (in seconds).
Audio Test

12. EXAMPLE 3
Model with a sine function
SOLUTION
STEP 1
Find the values of a and b in the model P = a sin bt. The maximum pressure is 2, so a = 2. You can use the frequency f to find b.
frequency =
period 1
2000 = b 2 π
= b Π
The pressure P as a function of time t is given by P = 2 sin 4000πt.

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

14. GUIDED PRACTICE
for Examples 2 and 3
Graph the function.
5. y = sin πx 1 4
SOLUTION
The amplitude is a = 1 4
and the period is 2 b π = 2.
Intercepts:
= (1, 0) ;
(2, 0)
(0 , 0); 1 2
2, 0 ( )
Maximums: 1 4 2, ( )
1 , 2 =
Minimum: 3 4 2, 1 ( ) – =
3 , 2

15. GUIDED PRACTICE
for Examples 2 and 3
Graph the function.
6. y = cos πx 1 3
SOLUTION
The amplitude is a =
and the period is 2 π = b 2. 1 3 = 1 2 , ( ); 3 4
2, 0 )
Intercepts:
Maximums: 1 3 2, ) ( 0, );
Minimum: 1 2 2, 3 ( ) – = 1,

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

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

18. GUIDED PRACTICE
for Examples 2 and 3 9.
What If ? In Example 3, how would the function change if the audiometer produced a pure tone with a frequency of 1000 hertz?
SOLUTION
STEP 1
Find the values of a and b in the model P = a sin bt. The maximum pressure is 2, so a = 2. You can use the frequency f to find b.
frequency =
period 1
1000 = b 2 π
= b
The pressure P as a function of time t is given by P = 2 sin 2000πt.

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

20. EXAMPLE 4
Graph a tangent function
Graph one period of the function y = 2 tan 3x.
SOLUTION b π = 3 .
The period is
Intercepts:
(0, 0)
Asymptotes:
x = 2b π =
2 3
, or x = ; 6
x = 2b π – =
2 3
, or x = ; 6

21. EXAMPLE 4
Graph a tangent function
Halfway points:
( , a) 4b π =
4 3
( , 2) 12
( , 2);
( , – a) 4b π – =
4 3
( , – 2) 12
( , – 2)

22. GUIDED PRACTICE
for Example 4
Graph one period of the function.
y = 3 tan x
SOLUTION

23. GUIDED PRACTICE
for Example 4
Graph one period of the function.
y = tan 2x
SOLUTION

24. GUIDED PRACTICE
for Example 4
Graph one period of the function.
f (x) = 2 tan 4x
SOLUTION

25. GUIDED PRACTICE
for Example 4
Graph one period of the function.
g(x) = 5 tan πx
SOLUTION