Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
Name
Key
Trigonometry: Modeling the Seas
ONE MORE LOOK!
On this worksheet, do as many problems as possible without your calculator. Find the exact
value or solution whenever possible. Otherwise, round your answer to the nearest hundredth.
1.
Find the value of:
 
3
a.
sin   
3  2
b.
27 
1
1

sec
 
 4  cos  274  cos  34 
 2
c.
2.
 3 
2
cos   
 4 
2
Solve for x within the given domain.
 1

cos  3x    , x - ,  
a.

4 2

3x  4   3  2k
3x 
x
7
12
 2k
7  24 k
36
or
d.
 117 
 
tan 
  tan    undefined
 6 
 2
b.
tan 2 (2x)  3, x 2 , 3 
tan  2 x    3
3x   12  2k
x
2 x   3  k
  24 k
36
x
25 17
for x    ,   , x  
,
,
36
36
 7 23 31
 ,
,
,
36 36 36 36
c.
6cos2 ( x)  5sin( x)  7, x 
   3 k
6
for x   2 ,3  ,
13 14 16 17
,
,
,
6
6
6
6
–3
tan(x)  4 cot(x)  3,
x0
2
tan x  tan4 x  3
x
d.
6 1  sin 2 x   5sin x  7
tan 2 x  4  3 tan x
0  6sin 2 x  5sin x  1
tan 2 x  3 tan x  4  0
0   3sin x  1 2sin x  1
 tan x  4  tan x  1  0
sin x   13
x  0.34  2k
or x  3.48  2k
or
tan x  1
tan x  4
or
x   4  k
x  1.33  k
sin x   12
7
x
 2 k
6

or x    2k
6
for x    32 , 0    4.71, 0  ,
x   54 , 1.82,  4
k
© 2005 Illinois Mathematics and Science Academy®
Trig. 18.1
Rev. S05
Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
4

3.
4
and
5
4
cos() 
5
If cos() 
a.
4.

Name

   0 , find:
2
(Note:  is a capital omega.)
3
4
c.
tan()  
e.

 4
sin    
2
 5
Key
3
5
4
5
b.
cos(  )  
d.

 4
sin    
2
 5
f.

5763
3
cos 
  

2 
5
Find equations of functions, using both the sine and cosine, which satisfies the given data:
 
 5 
,1
a.
maximum at  , 5  and the next minimum at 
4 
 6 
5

7
7
2
1
12
2 period  6  4  12  period  6 ; B  period  7
 12 
 
y  2sin   x     3
24  
7
 12 
 
y  2cos   x     3
4 
7
  1
period is 8π, range is 4, 2 , and contains the point  ,  .
 2 2
b.
y  3sin  14  x  c1    1 or y  3cos  14  x  c2    1
1
2
 sin  14  2  c1    sin


8

 c41  c1  4  sin 1  12   8  
6
or c2  4  cos 1  12   8  
1
5
y  3cos   x 
6
4
1
 
y  3sin   x     1
6 
4
c.
with the graph below
5
6

  1

amplitude  12 1  9  5
1
5

5
10
vertical shift  12 1  9  4
5
2
B
4
2
4
 2
 
y  5sin  x   4
2 


y  5cos   x  1   4
2

9
© 2005 Illinois Mathematics and Science Academy®
period  10  period  4
Trig. 18.2
Rev. S05
Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
Name
5.
Key
Find the equation of a cosine function whose graph is the same as the graph of

 
 

y  2 sin  4 x    7 .  2sin  4  x     7

3
  12  
period  24  2 , cosine is a shift of 14 of a period left of sine so phase shift  12  8  524
 
5  
y  2 cos  4  x 
 7
24  
 
6.
How many feet above the
plain is the top of the
mountain, given   9
and   24 .
tan 24  hx  h  x tan 24
h
24
tan 9  x 5280
 xxtan
5280
x tan9  5280tan9  x tan 24
h
 24
 9
x
 5280 feet
5280 tan 9  x  tan 24  tan 9   x 
5280 tan 9
tan 24  tan 9
5280 tan 9 tan 24
 1298 feet
tan 24  tan 9
The table below gives the latitude that the sun is directly above at various times during
the year. Find the equation of a function that gives the latitude of the sun of any day of
the year. Let January 1 = 1 and December 31 = 365 (ignore leap year). Note that your
function may only approximate the data.
h  x tan 24 
7.
Day of Year
Day of Month
Latitude
21
Jan. 21
-20.00
52
Feb. 21
-10.72
80
Mar. 21
0.05
111
Apr. 21
11.70
141
May 21
20.08
172
Jun. 21
23.33
Day of Year
Day of Month
Latitude
202
Jul. 21
20.55
233
Aug. 21
12.27
264
Sep. 21
1.27
294
Oct. 21
-10.53
325
Nov. 21
-19.82
355
Dec. 21
-23.43
Do a stat plot and SinReg on the calculator:
y  23.9sin  0.02 x 1.30  0.60  23.9sin  0.02  x  65   0.60
© 2005 Illinois Mathematics and Science Academy®
Trig. 18.3
Rev. S05
Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
Name
Key
Solve each equation. Find exact solutions whenever possible and give approximations to the
nearest hundredth of a radian as necessary.
3cot 2 (x)  5 cot(x)  2  0
8.
3sec2 (x)  5sec(x)  12  0
9.
 3sec x  4  sec x  3  0
 3cot x  1 cot x  2   0
cot x  13
tan x  3
x  1.25  k
10.
or
cot x  2
sec x   43
tan x  
cos x  
1
2
or x  0.46  k
k
tan 2 (x)  3cot 2 (x)  4
2
2
cos 2 x
sin 2 x
tan x  1
tan x  1
x

4
 k
0  2sin 2 x  sin x  1
tan x  3
or
0   2sin x  1 sin x  1
tan x   3
x

3
 sin x  0 
1  sin 2 x  sin x  sin 2 x
2
or

 sin1 x  1 sin 2 x
cos 2 x  sin x  sin 2 x
x  1 tan 2 x  3  0
2
x  1.23  2k
cot 2 (x)  csc(x)  1
tan 4 x  4 tan 2 x  3  0
 tan
or
cos x  13
k

tan x  3  4 tan x
4
x  2.42  2k
11.
tan 2 x  tan32 x  4
sec x  3
or
3
4
 k
sin x 
x
k

6
or x 
© 2005 Illinois Mathematics and Science Academy®
Trig. 18.4
1
2
or
sin x  1
 2 k
or
5
 2 k
6
k
x
3
 2 k
2
Rev. S05