Download PRECALCULUS A TEST #5 – TRIGONOMETRIC FUNCTIONS

PRECALCULUS A
TEST #5 – TRIGONOMETRIC FUNCTIONS, PRACTICE
APPENDIX A – Equations and Inequalities & APPENDIX B – Graphs of Equations
1)
Express in interval notation: x | x  3.
2)
Express in set notation:  ,  2 .
3)
A painter brings a 20 foot ladder to paint a house. When the ladder is leaned against the house,
it reaches a point 19.1 feet above the ground. How far from the house is the end of the ladder?
Round the answer to the nearest tenth. HINT – Drawing a picture would be beneficial.
4)
Point P has coordinates (–4, 3) and point Q has coordinates (2, –5). Find the coordinates of the
midpoint of PQ.
5)
Point P has coordinates (–2, –1) and Point Q has coordinates (1, 5). Find the distance between
these points. Express the answer as a radical in simplest form and as a decimal, rounded to the
nearest tenth.
SECTION 1.1 – Angles
6)
Evaluate 13° 24' + 38° 45'.
7)
One angle of a triangle has measure 38° 23' and another angle measures 67° 48'. Find the
measure of the third angle.
8)
Convert 38° 23' 17" to decimal degrees. Round the answer to three decimal places.
9)
Convert 59.0854° to degrees, minutes, and seconds.
10)
An angle has a measure of –72°. Find the measure of the smallest positive angle that is
coterminal with this angle.
11)
An angle has a measure of 755°. Find the measure of the smallest positive angle that is
coterminal with this angle.
12)
An angle has a measure of 216°. Find the measure of two other angles, one positive and one
negative, which are coterminal with this angle, and indicate the quadrant where each angle is
located.
13)
An airplane propeller rotates 850 times per minute. Find the number of degrees that a point on
the edge of the propeller will rotate in ½ second.
SECTION 1.2 – Angle Relationships and Similar Triangles
14)
A 90-foot-tall building casts a shadow of 22 feet at the same time a tree casts a shadow of 8
feet. To the nearest tenth of a foot, how tall is the tree?
15)
A flagpole casts a shadow that is 50 feet long. At the same time, a woman standing nearby who
is five feet four inches tall casts a shadow that is 40 inches long. How tall is the flagpole?
16)
Ruby is standing in her back yard and she decides to estimate the height of a tree. She stands so
that the tip of her shadow coincides with the tip of the tree’s shadow. Ruby is 66 inches tall.
The distance from the tree to Ruby is 95 feet and the distance between the tip of the shadows
and Ruby is 7 feet. How tall is the tree, measured to the nearest tenth of a foot?
HINT – Drawing a picture would be beneficial.
SECTION 1.3 – Trigonometric Functions
17)
The terminal side of an angle θ in standard position passes through the point (8, –15). Find the
values of the six trigonometric functions.
18)
The terminal side of an angle θ in standard position passes through the point (–1, 6). Find the
values of the six trigonometric functions. Make sure all answers are expressed in proper radical
form.
19)
The terminal side of an angle θ in standard position passes through the point (–2, –6). Find the
values of the six trigonometric functions. Make sure all answers are expressed in proper radical
form.
20)
Evaluate 4sin90  2cos270  3tan 0.
21)
Evaluate csc2 90   sin 270 tan180.
SECTION 1.4 – Using the Definitions of the Trigonometric Functions
22)
1
If cot    , find tan  .
3
24)
If cos  
25)
If tan  
26)
If csc  
23)
2
If sin   , find csc .
5
3
and  is in Quadrant II, find sin  and tan .
4
4
and  is in Quadrant III, find sin  and cos .
3
8
and  is in Quadrant IV, find the value of the other trigonometric functions.
5
27)
Identify which of the following statements are impossible:
A) sin   1.5
B)
cos  0.92
C) tan   3.71
D)
sec  0.5
E) csc  1.2
F)
cot   0
**********************ANSWERS**********************
1)
2)
x | x  3  All numbers greater than or equal to 3 
ANSWER  3,    Must use bracket on left side because of
 , 2  All numbers less than

2
Must use < because of parentheses on right  ANSWER  x | x  2
3)
See diagram to right  x2  19.12  202  x2  364.81  400 
x2  35.19  x2  35.19  x  5.932116  x  5.9 feet
4)
 4  2 3   5   2 2 
Midpoint  
,
 ,
  Midpoint   1,  1
2   2
2 
 2
20
x
5)
Distance 
1   2  5   1
2
2
 32  62  9  36  45  9  5 
Distance  3 5  6.7082039  6.7
6)
13 24' 38 45'  51 69'   51  1   69  60  '  52 09'
7)
38 23' 67 48'  105 71'  105  1   71  60  '  106 11' 
180  106 11'  179 60' 106 11'  73 49'
8)
38 23' 17"  38  23/60  17/3600  38.388056  38.388
9)
59.0854  0.0854  60  5.124'  0.124  60  7.44"  59 05' 7.44"
10)
72  360  288
12)
216  360  576; 216  360  144  Both angles are in Quadrant III
13)
850  360  306,000 per 60 seconds  306,000  60  5,100 per second 
1
1
5,100   2,550 per second
2
2
11)
755  360  395  360  35
19.1
14)
8
x
15)


22 90
Cross multiply  22 x  720  x  32.7 feet
See diagram below 
See diagram below 
40 in 64 in


50 ft
x
Cross multiply  40 x  3200  x  80 feet
x
90
x
64 in
8
16)
22
50 ft
See diagram below 
40 i n
7
5.5

 Cross multiply  7 x  561  x  80.1 feet
102
x
x
5.5 ft
95 ft
7 ft
102 ft
17)
r  x2  y 2  r  82   15  r  64  225  r  289  r  17
2
y
15
15
 sin  

r
17
17
y
15
15
tan    tan  

x
8
8
r
17
sec   sec 
x
8
sin  
x
8
 cos 
r
17
r
17
17
csc   csc 

y
15
15
x
8
8
cot    cot  

y
15
15
cos 
18)
r  x2  y 2  r 
sin  
19)
 1
2
 62  r  1  36  r  37
y
6
6  37
6 37
 sin  


r
37
37
37  37
cos  
x
1
1 37
 37
 cos  


r
37
37
37  37
tan  
y
6
 tan  
 6
x
1
csc 
r
37
 csc 
y
6
sec 
r
37
 sec 
  37
x
1
cot  
x
1
1
 cot  

y
6
6
r  x2  y 2  r 
sin  
 2
2
  6   r  4  36  r  40  4  10  r  2 10
2
y
6
3
3  10
3 10
3 10
 sin  



 sin   
r
10
10
2 10
10
10  10
x
2
 cos 

r
2 10
y
6
tan    tan  
3
x
2
r
2 10
csc   csc 

y
6
cos 
1
1  10
 10
10


 cos  
10
10
10
10  10
10
10
 csc  
3
3
r
2 10
10
 sec 

 sec   10
x
2
1
x
2 1
cot    cot  

y
6 3
sec 
20)
sin90  1, cos270  0, tan 0  0  4sin90  2cos270  3tan 0 
4 1  2  0  3  0  4  0  0  4
21)
csc90  1, sin 270  1, tan180  0  csc 2 90  sin 270 tan180  
12   1  0  1  0  1
22)
cot  and tan  are reciprocals  tan   3
23)
sin  and csc are reciprocals  csc 
5
2
2
24)

3
3
3
2
Use sin   cos   1 to find sin   cos  
 sin 2    
  1  sin    1 
4
4
16


2
sin 2  
2
13
13
13
 sin   
 sin   

16
16
4
13
4
sin 
13/4
13
tan  

 Multiply top & bottom by 4 

cos  3/4
 3
Since  is in Quadrant II, sin  is positive  sin 
Multiply top & bottom by 3 
13  3
39
39

 tan   

3
3
 3 3
2
25)
Use tan 2   1  sec2  to find sec  tan  
4 4
16
    1  sec2    1  sec2  
3 3
9
25
25
5
 sec2   
 sec    sec 
9
9
3
Since  is in Quadrant III, sec is negative  sec  
sec and cos are reciprocals  cos   
5
3
3
5
2
3
 3
Use sin 2   cos2   1 to find sin   cos    sin 2       1 
5
 5
9
16
16
4
 1  sin 2  
 sin   
 sin    
25
25
25
5
4
Since  is in Quadrant III, sin  is negative  sin   
5
sin 2  
5
8
2
5  5
Use sin 2   cos2   1 to find sin   sin         cos2   1 
8  8
26)
csc and sin  are reciprocals  sin   
25
39
39
39
 cos2   1  cos2  
 cos  
 cos  

64
64
64
8
39
Since  is in Quadrant IV, cos is positive  cos 
8
8
8  39
8 39
cos and sec are reciprocals  sec 

 sec 
39
39
39  39
sin 
5/8
5
tan  

 Multiply top & bottom by 8 

cos
39/8
39
5  39
5 39
5 39
 tan  

39
39
39  39
39
tan  and cot  are reciprocals  cot   
 Multiply top & bottom by 39 
5 39
Multiply top & bottom by 39 

27)
39  39
39 39
39
39


 cot   
5  39
5
5
5 39  39
1  sin  1;  1  cos  1  sin  and cos must fall between  1 and 1, inclusive
sec  1 or sec  1; csc  1 or csc  1  sec and csc are NEVER between  1 and 1
tan  and cot  may be any real number
sin   1.5  Impossible, because sin must fall between  1 and 1
sec  0.5  Impossible, because sec may never be between  1 and 1