Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
Name
Key
Trigonometry: Modeling the Seas
PRACTICE 2
1.
 x  2 
 1 . Find each of the following.
Let f (x)  4 sin 
 3 
a. amplitude
b. phase shift
c. period
4
2
6
d. maxima
(both coordinates)
x  2
3
e. minima
(both coordinates)
 2  2k
x  2
3
x  2  32  6k
x

2
 6k , k 
 32  2k
x  2  92  6k
and y  3
x
5
 6 k , k 
2
and y  5
f. Sketch the graph of ƒ. (Mark the scales carefully.)
3
 72
2
 2

5
2
4
5
2.
Find the exact value of each of the following.
 5 
 7  2 2 3
b. tan     3


a. sec 

 3
 6 
3
3
 3 
d. csc    1
 2 
1
 5 

e. sin 

 6 
2
© 2005 Illinois Mathematics and Science Academy®
Trig. 12.1
 9 
 1
c. cot 
 4 
 7 
f. sec 
 2
 4 
Rev. S05
Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
Name
3.
5
Key
Write two different equations of sinusoidal curves satisfying the following information:
one max is at (/4, 5) and the next min is at (/2, 1)
[Answers may vary!]


1
2 period  4  period  2  B  4
Amplitude  12  5 1  2
Vert. shift  12 5  1  3
1
4.



8
4
2
 
 
y  2sin  4  x     3 or y  2cos  4 x   3
8 
 
Solve, finding all solutions.


b. 2 cos  x    1

6
a. tan(3x)  1  0
tan  3 x   1
cos  x  6   12
3 x   4  k
x  6   3  2k
x

12

k
,k
3
x
3x  
x
5.
3
4

4
1
2
 2 k

2 k
, k
3
2
 2 k
x
or

6
 2 k , k 
3
 x  3 
d. sin 


 4 
2
c. sec(3x)   2
cos  3x   

x 3
4
 3  2k
x  3 
x
4
3
x 3
4
 8k
13
 8k
3
or

2
3
x  3 
x
 2 k
8
3
 8k
17
 8k ,
3
k
Sketch the following graphs.
 x
b. h(x)  3sec  
 2
a. g(x)  tan(2x)


4

4
© 2005 Illinois Mathematics and Science Academy®
3
Trig. 12.2


3
Rev. S05
Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
Name
6.
State the domain and range of each function.
a. f (x)  4 sin 5x   3
b. g(x)  4 tan(2x)
 k


Dg :  x : x  
,k 
4 2


Rg :
Df :
R f :  y : 7  y  1
7.
Key


c. h(x)  2 cot  x  

3
 x
d. j(x)  3sec  
 2



Dh :  x : x   k , k  
3


Rh :
D j :  x : x    2k , k 

R j :  y : y  3 or y  3
Find the zeros of the following functions.
a. f (x)  4 sin 3x   


b. g(x)  tan  x  

6
3 x    k
  k
x
,k
3
x  6  k
x

6
 k , k 


c. h(x)  2 cos  x    1

8
cos  x  8   12
x  8   3  2k
x
8.
5
 2 k
24
or
x
11
 2 k , k 
24
Find an equation for the sinusoidal graph.
 
2  
y  5cos  2  x 
 3
3  
 
[Other answers possible!]

2
 12 period
   period
2B
© 2005 Illinois Mathematics and Science Academy®
Trig. 12.3
Rev. S05