Download Name: Math 2412 Activity 4(Due by May 2) 1. Use properties of

Name:_____________________ Math 2412 Activity 4(Due by May 2)
1. Use properties of similar triangles to find the values of x and y.
x  2y
7
74
x5
x y
14
74
21
2. For the angle  in standard position with the point  5,12  on its terminal side, find the
values of the six trigonometric functions:
sin  
cos 
tan  
csc 
sec 
cot  
 5,12

3. Find one solution of the equation sin  2  10   cos  3  10  . {Hint: cos x  sin  90  x  .}
4. Find all the trigonometric function values of  , if csc  2 and  is in Quadrant III.
sin 
cos 
tan  
csc  2
sec 
cot  
5. Find the exact value of each labeled part:
a
m
q
45
n
a
n
60
q
6. Find all the exact trigonometric function values of 1590 .
sin1590 
cos1590 
tan1590 
csc1590 
m
7
sec1590 
cot1590 
7. Solve the right triangle to the nearest tenth of a degree and tenth of a foot:
A
89.5 ft.
b
47.9
m  A 
B
a
C
a
b
8. Solve the right triangle to the nearest degree and the nearest foot:
A
c
137 ft.
C
156 ft.
B
9. Find h to the nearest tenth.
h
35
21
x
135
x
x  135 x 135
{Hint: cot  35   and cot  21  
.}
 
h
h
h
h
10. Find h to the nearest tenth.
135
35
21
x
{Hint: cot  35  
h
x
xh
x
h
and cot  21  
.}


135
135 135 135
11. Find the area of the indicated sector:
5

8
12. Find the measure of the central angle,  , in radians.
5

20
13. The rotation of the larger wheel causes the smaller wheel to rotate. Find the radius of the
larger wheel if the smaller wheel rotates 90 when the larger wheel rotates 60 .
12 ft.
14. Graph the function y  2cos  x  on the interval 0,2 .
r
15. Graph the function y  32 sin  32 x  on the interval  0,3  .
3
4
3
9
4
3
2
16. Graph the function y  3sin  12 x   3 on the interval 0,4  .

2
3
4
5
17. Graph the function y  3cos  4 x     1 on the interval   4 , 4  .


4


8

8

4
1
18. Graph the function y   sin  34 x  8   2 on the interval
2

6
5
6
3
2
 6 , 176  .
13
6
17
6
19. Graph the function y  3sec  2 x  3  on the interval

6
5
12
 6 , 76  .
7
6
11
12
2
3
20. Graph the function y  2csc  x  2   1 on the interval   2 , 32  .


2

2

3
2
1
21. Graph the function y  sec  x  2   2 on the interval  2,4 .
2
x 
22. Graph the function y  tan    on the interval
2 4

2

3
2
 2 , 52  .
2
5
2
3
23. Graph the function y  3cot  4 x     2 on the interval   4 ,0 .


4

3
16


8


16
24. Graph the function y  4tan  3 x     2 on the interval 3,6 .
25. Determine the range of the following functions:
a) y  3sin  2 x   7
b) y  2sec  2 x  11  8
26. Verify the identity cos4 x  sin 4 x  2cos2 x  1
tan 2 x  sec2 x
sec x
27. Verify the identity
.

2
2
cos x  sin x sec x  2cos x
28. Show that the equation cos2 x  cos x  sin x is not an identity by demonstrating that for a
specific value of x it is false.
29. Show that the equation sin 2 x  sin x  cos x  1 is not an identity by demonstrating that for a
specific value of x it is false.
30. Find the exact value of cos 165  .
31. Find an exact value of  that makes cot   10   tan  2  20
32. Verify the identity cos  x  90   sin x  sin 2 x  1  cos 2 x .
33. Find the exact value of cos 14  cos  29   sin 14  sin  29  .
34. Find the exact value of
tan 512  tan 4
.
1  tan 512 tan 4
35. Find the exact value of sin165 .
36. Verify the identity tan  x  y   tan  y  x  
37. Verify the identity
sin  x  y  cot x  cot y

.
cos  x  y  1  cot x cot y
38. Find the exact value of cos2 12  sin 2 12 .
39. Find the exact value of 4sin 22.5 cos 22.5 .
40. Verify the identity
2  tan x  tan y 
.
1  tan x tan y
1  cos 2 x
 cot x .
sin 2 x
 true.
1  tan 2 x
41. Verify the identity
 cos2 x .
1  tan 2 x
42. Find the exact value of cot 2 , if tan   
5
and 90    180 .
2
43. Verify the identity tan 2x  csc x  cot x .
1  tan 2 2x
44. Verify the identity cos x 
.
1  tan 2 2x

2
45. Find the exact value of sin 1  
.
2


 2 
46. Find the exact value of sec1  

 3

 1 
47. Find the exact value of tan  2cos 1    .
 4 


 3
 5 
48. Find the exact value of cos sin 1     cos 1     .
 5
 13  


 1 
49. Find the exact value of sin  2sin 1     .
 3 

50. Solve the equation cos2   cos  2  0 on the interval  0,2  .
51. Solve the equation 4sin 2   1  0 on the interval  0,2  .
52. Solve the equation sec2   tan   1 on the interval  0,2  .
53. Solve the equation cos2   sin  2 on the interval  0,2  .
54. Solve the equation cos 2x  0 on the interval  0,2  .
55. Solve the equation cos 2 x  sin 2 x  0 on the interval  0,2  .
56. Solve the equation tan 2x  sin x on the interval  0,2  .
57. Solve the equation sin x cos x 
1
on the interval  0,2  .
2
Sketch the solutions of the following polar coordinate equations.
58. r  1  sin 
59. r  1  2cos
Find the points of intersection of the solution curves of the following pairs of polar coordinate
equations.
60. r  1  cos , r   cos
61. r  2cos3 , r  1
Find the points of intersection of the curves defined by the following parametric equations.
x 1 t
x 1 s
; 3  t  2
62.
; 3  s  2
and
y  2  2s
y  t2 1
x  sec s
x  2cos t
;0  t  2
63.
y  3sin t
;  3  s  3
and
y  tan s
x  cos s
x  cos t
;0  t  2
64.
y  sin 2t
;0  s  2
and
y  12 sin s
65. Find the exact value of each part labeled with a variable.
8
30
x
y
w
60
z
66. The tires of a bicycle have a radius of 1.25 ft, and are turning at the rate of 5 revolutions per
second. How fast is the bicycle traveling in feet per second?
67. If tan x  .75 and cos x  .8 , then find the value of tan   x   cos   x  .
 
68. Find the exact value of cos   .
 12 
{Hint:

12


3


4
and cos  A  B   cos A cos B  cos A cos B .}
 5 
69. Find the exact value of tan   .
 12 
{Hint:
70. Find the exact value of cos
5  
tan A  tan B
.}
  and tan  A  B  
12 6 4
1  tan A tan B
11
.
12
11
11
A
1  cos A
 6 .}
{Hint: cos  
and
12
2
2
2
Find the exact value of the following:

 1 
71. sin  sin 1   
 12  

  4  
72. sin 1  sin 

  3 

 2 
73. cos  sin 1   
 3 

74. sin  tan 1  2  

 1 
75. tan  cos 1    
 4 

For each of the following, find sin  x  y  , cos  x  y  , tan  x  y  , and the quadrant of x  y .
76. sin x 
1
4
, cos y  , x in quadrant I, y in quadrant IV
10
5
2
1
77. sin y   , cos x   , x in quadrant II, y in quadrant III
3
5
Find the sine and cosine of the following
1
78. B , given cos 2 B  , B in quadrant IV
8
5
79. 2y , given sec y   , sin y  0
3
Find the following:
3
 A
80. sin   , given cos A   , with 90  A  180 b) sin 2x , given sin x  .6 , with
4
2

2
 x 

1
81. sin y , given cos 2 y   , with  y  
3
2
Exactly solve the following trigonometric equations on the interval  0,2  .
 x
 x
85. csc    sin  
3
3
84. sec 4  2 x   4
82. sin 2 x  1
83. 3cos2 x  2cos x  1  0
86. sin x  sin 2 x
87. cos 2 x  cos x  0
x
90. cos  1
2
91.  sin x  2   cos4 x  1
88. sin 2 x  2cos 2 x
89.
92. 6sin 2 x  17sin x  12  0
6
93. Sketch the graph of the solution to the polar coordinate equation r  sin 2 .
1
r

4
1

2

3
4
2 sin3x  1  0
5
4
3
2
2
7
4

94. Sketch the graph of the solution to the polar coordinate equation r  1  cos .
r
2
1

2

3
2
2

95. Find the points of intersection of the solution curves of the polar coordinate equations
r  2  cos2 and r  2  sin  .
96. Find the points of intersection of the solution curves of the polar coordinate equations
r  2sin and r  sin   cos .
97. Graph the function y  tan   x   1 on the interval   2 , 2  .
98. Graph the function y   sin  2 x  on the interval  0,   .
99. Determine the range of the function y  8sin  5x     7 .
98. If cos x  13 , then find the exact value of sin   x  tan   x   sin   x  cot   x  .
Find the exact value of the following.

 4 
100. sin  2cos 1   
 5 

{Hint: sin 2 A  2sin A cos A .}

 1
 2 
101. sin sin 1     sin 1   
 4
 3 

{Hint: sin  A  B   sin A cos B  cos A sin B .}

 1 
102. tan  12 sin 1    
 3 

{Hint: tan
A
sin A
1  cos A
.}


2 1  cos A
sin A
103. cos  12 sin 1   14  
104. Sketch the graph of the solution to the polar coordinate equation r  cos 2 .
r
1

4
1

2
3
4

5
4
3
2
7
4
2

105. Sketch the graph of the solution to the polar coordinate equation r  1  2sin  .
r
3
1

2

7
6
3
2
11
6
2

1
106. Find the points of intersection of the solution curves of the polar coordinate equations
r  1  sin  and r  3sin  .
107. Find the points of intersection of the solution curves of the polar coordinate equations
r  2sin 2 and r  1.
108. Find the area of the region that is inside the solution curve of r  2sin but outside the
solution curve of r  sin .
109. Given that a  4i  3 j and b  2i  j and another vector r  6i  7 j , find numbers k and
m so that r  ka  mb .
110. Express c in terms of a and b , given that the tip of c bisects the line segment.

b

a
111. For what values of x are xi  11 j and 2xi  xj orthogonal?

c
112. Given that a  i  xj  k and b  2i  j  yk , find all values of x and y so that a  b and
a  b .
113. Use the dot-product to show that an angle inscribed in a semi-circle is a right angle.



(Look at a  b  a  b .)
a b
a
b
a b
b
114. Show that the sum of the squares of the lengths of the diagonals of a parallelogram equals
the sum of the squares of the lengths of the four sides.
a
2
Expand a  b  a  b
2
a b
by using the dot-product.
b
b
a b
a
115. It looks as if a  b and a  b are orthogonal. Is this mere coincidence, or are there
circumstances where we would expect the sum and difference of two vectors to be



orthogonal? Find out by expanding a  b  a  b  0 .
a b
b
a
a b
b
116. Given vectors a and b , let m  a and n  b , show that
a) na  mb and na  mb are orthogonal.
b) c  na  mb bisects the angle between a and b .
117. Find all vectors v in the plane so that v  1 and v  i  1 .
Graph each parabola.
118. x 2  4 y
119.  y  1  4  x  2 
2
120. x2  8x  8 y
Graph each ellipse.
x2 y 2
121.

1
25 16
4 x 2 16 y 2
122.

1
81
25
123. 6 x 2  5 y 2  30
124. 9x2  18x  4 y 2  8 y  23  0 .
Graph each hyperbola.
x2 y 2
125.

1
16 25
126. 4 y 2  x 2  1
127. 4 x 2  25 y 2  100
128. 9 x2  18x  4 y 2  8 y  31  0 .
129. Find an equation for the parabola with focus of  4,4  and directrix of y  2 .
130. Find an equation of the hyperbola satisfying the given conditions:
Endpoints of transverse axis:  4,0  ,  4,0  ; asymptote y  2 x
131. Solve the system
x2  y 2  9
x  y 9
2
2
.
Solve the following systems of equations. Check to see if your answer agrees with the graph.
132.
x  y  1
(line)
y  x 2  1 (parabola)
133.
x 2  y 2  5 (circle)
3x  y  5 (line)
134.
136.
4 x 2  y 2  4 (hyperbola)
4 x 2  y 2  4 (ellipse)
y  x 2  2 x  1 (parabola)
y  1  x 2 (parabola)
135.
3x 2  4 y 2  16 (ellipse)
2 x2  3 y 2  5
137.
(hyperbola)
y  x 2  2 (parabola)
x 2  4 y 2  16
(ellipse)
138. Find the values of x and y in the figure.
x
10
y
17
9