Download MATH-1330

MATH-1330
Exam 2
Spring 2013
Name ___________SOLUTIONS____________
Rocket Number __________________________
INSTRUCTIONS: You must show enough work to justify your answer on ALL problems except for Problems
5a, 5b, 5c, 5d, and 5e. Correct answers with no work (or inconsistent work) shown will not receive full credit.
All answers are to be exact; no decimal approximations. You are NOT allowed to any electronic device for
this exam.
1.
Given the triangle below, find x. Set up an equation and solve. (5 pts.)
38
x
 tan 26.3  x  38 tan 26.3 
38
26.3 
x
x  38 tan 26.3 
2.
Find the exact angle  , where 0    360 , if the terminal side of  passes through the point
(  16 , 12 ) . (5 pts.)
The point (  16 , 12 ) is in the second quadrant. Thus, (the terminal side of)  is the second quadrant.
tan  
y
12
3
 
 
x
16
4
  180    '  180   tan  1
  180   tan  1
3.
tan   
3
3
3
 tan  ' 
  '  tan  1
4
4
4
3
4
3
4
From the top of a 60-meter building, the angle of depression to an object on level ground below is 18  .
How far is the object from the top of the building? Draw a picture and label known information. Indicate
any variable you use. Set up an equation and solve. (6 pts.)
---------------------18 
z
 csc 18  z  60 csc 18 
60
z
60 m
OR 60  sin 18   60  z sin 18   z  60
z
sin 18 
18  Object
Answer: 60 csc 18  m OR x 
60
m
sin 18 
4.
A 35-foot ladder is leaning against a vertical wall. If the bottom of the ladder is 15 feet from the base of the
wall, then find the angle of elevation of the ladder. Draw a picture and label known information. Indicate
any variable you use. Set up an equation and solve. (5 pts.)
cos  
15
3
3

   cos  1
35
7
7
35 ft

15 ft
Answer: cos  1
5.
3
7
Find the exact value of each of the following. (3 pts. each)
1


2
6
a.
Arc sin
b.
Arc cos (  1)  
c.

3
  5
cos  1  

2 
6

d.
tan  1 ( 
e.
sin
1
3)  
NOTE: sin

6

1

   
  , 
and
2
6
 2 2
NOTE: cos    1 and   [ 0 ,  ]

3

2 

  

2 
4

NOTE: cos
3
5

5
 [ 0,  ]
  cos
 
and
6
6
2
6

 
 
NOTE: tan      tan
3
 3
3 and 

   
 ,

3
 2 2 
2


   
 
NOTE: sin      sin
and     , 
 
4
4
2
 2 2
 4
6.
Sketch two cycles of the graph of the following function. Sketch two cycles on one side of the y-axis.
Label the numbers on the x- and y-axes as needed. Label where the cycles begin and end. Then label the
numbers between these numbers. (8 pts.)
y  cos
Period =
3x
4
Amplitude: 1
Period: 8 
3
2
4
8
 2  
3
3
3
4
Phase Shift: None
1
1 8
1 2
2
period = 
 

4
4 3
1 3
3
y
1
2
3
4
3
6
3
||
8
3
12 12
 
3 3
|| ||
14 
3
16 
3
x
4 4 
2
1
10 
3
6
3
||
2
7.
Sketch two cycles of the graph of the following function. Label the numbers on the x- and y-axes as needed.
Label where the cycles begin and end. Then label the numbers between these numbers. (6 pts.)
y 
1
tan 6  x
4
Amplitude: None
Period: 1
6
y
Phase Shift: None

1
12
1
12
1
6
3 1

12 4
x
1
1
1
2
3
1





12
6
12
12
12
4
8.
Sketch two cycles of the graph of the following function on the same side of the y-axis. Label the numbers
on the x- and y-axes as needed. Label only where the cycles begin and end. Do NOT label the numbers
between these numbers. (8 pts.)
y
5 

3 csc  2 x 

6 

y
 
5
3 sin 2  x 
12
 
Amplitude: None
Period: 



Phase Shift:
5
units left
12
y
3

3
19 
12
7
12
31
12
x

9.

Find the exact value of cot sin

1
5
5
12 
7
   


12
12
12
12
 4 
   (8 pts.)
 7 
4
4
 4
 sin  ' 
Let   sin  1    . Then sin   
7
7
 7
7
4
'

 4 
tan sin  1     tan    tan  '  
 7 

33
sin
1
 4
   is in the ___IV__ quadrant
 7

cot sin

1
 4 
   = 
 7 
33
4
4
33
10.
Use one Pythagorean Identity and Basic Identities to find the exact value of sin  and sec  if
cot   
y
5 and sin   0 . (10 pts.)
y
C
S
S
x
 is in II
x
C
csc 2   cot 2   1  csc 2   5  1  csc 2   6  csc   
 in II  csc  
cot  
6  sin  
11.
1
6
cos 
 cos   cot  sin   
sin 
 is in the ____II___ quadrant
6
 1 
  
5
 6 


sin  =
5
6
1
6
sec  = 
Find all the exact solutions (in degrees) of the equation ( tan   1) (
Put a box around your answer(s).
6
5
3 csc   2 )  0 . (10 pts.)
tan   1  0  tan   1  tan  '  1   '  tan  1 1  45 
I:   45   n  360  , where n is an integer
3 csc   2  0  csc   
III:   240   n  360 
III:   225   n  360 
3
2
 sin   
 sin  ' 
2
3
IV:    60   n  360 
3
3
  '  sin  1
 60 
2
2
12.
Find the exact value of sin 285 . (6 pts.) Put a box around your answer.
sin 285   sin 75
sin 75  sin ( 30   45  )  sin 30  cos 45   cos 30  sin 45  
2

4
6

4
OR sin 75 
6 
4
2
1  cos 150 

2
sin 285   sin 75  
13.
sin 285   sin 75  

3

1  


2

 
2
2 
1
2
6 
4
3
2  2 
2
2
3
2


2
2
2
2 
3
4

2 
2
3
2
Find the exact value of sin 2  and sec 2  if sin  

2
1


2 2
3

    . (9 pts.)
and
2
6
      is in II
cos    cos  '  
6
3
33
6
'
33
 3 
 
sin 2   2 sin  cos   2 
 6 


cos 2   cos 2   sin 2  
sin 2   
11
6
33
6
  3 
  
 
  3 
 

33
3
30
5



36
36
36
6
sec 2  
6
5
33
6

99
3 11
  
 
 

18
18

11
6
3
14.
Find all the solutions (in radians) to the equation cos 2 x 
1
in the interval [   , 2  ] .
2
(6 pts.)
Put a box around your answer.
Let u  2 x .
cos 2 x 
u is in I or IV
cos u 
u 
I:
u 
x 


x 
 2 n  2 x 
3
( 6 n  1) 
6
u  
1
1
1

 cos u ' 
 u '  cos  1 
2
2
2
3
 2 n  , where n is an integer
3
u  
IV:
1
1
 cos u 
2
2

3

3
 2 n  x 
n  0: x 


6
 n 
, n  1: x 
6


6
7
6
6 n
6 n  

6
6
, n   1: x  
5
6
 2 n




6 n
6 n  
 2 n  2 x  
 2 n  x  
 n  


3
3
6
6
6
6
( 6 n  1) 
6
n  0: x  

6
, n  1: x 
5
6
, n  2: x 
11
6
FORMULAS
cos (    )  cos  cos   sin  sin 
cos (    )  cos  cos   sin  sin 
sin (    )  sin  cos   cos  sin 
sin (    )  sin  cos   cos  sin 
tan (    ) 
tan   tan 
1  tan  tan 
tan (    ) 
tan   tan 
1  tan  tan 
cos 2  cos 2   sin 2 
sin 2  2 sin  cos 
cos 2  2 cos 2   1
cos 2  1  2 sin 2 
cos

2

1  cos 
2
sin

2

1  cos 
2
tan

2

1  cos 
1  cos 