Math 2412 Activity 4(Due with Final Exam)
1. Use properties of similar triangles to find the values of x and y.
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
74
74
7
14
5x
21
2x y
x y
5,12
3. Find one solution of the equation sin 2 10 cos 3 10 . {Hint: cos sin 90x 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
n
q
6. Find all the exact trigonometric function values of 1590 .
sin1590
cos1590
tan1590
csc1590
45
60
a 7
m
n
q
sec1590
cot1590
7. Solve the right triangle to the nearest tenth of a degree and tenth of a foot:
m A b a
8. Solve the right triangle to the nearest degree and the nearest foot:
47.9
89.5 ft.
A
B C a
b
c
A
B C 156 ft.
137 ft.
9. Find h to the nearest tenth.
{Hint: cot 35x
h and
135 135cot 21
x x
h h h
.}
10. Find h to the nearest tenth.
{Hint: cot 35135
x and cot 21
135 135 135
x h x h .}
11. Find the area of the indicated sector:
21
135
h
35
x
21 35
x
135
h
8
5
12. Find the measure of the central angle, , in radians.
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 .
14. Graph the function 2cosy x on the interval 0,2 .
12 ft. r
20
5
4 3 2 5
15. Graph the function 3 22 3siny x on the interval 0,3 .
16. Graph the function 12
3sin 3y x on the interval 0,4 .
3 3
2
3
4
9
4
6
7
6
2
3
5
12
11
12
2
3
2
2
19. Graph the function 33sec 2y x on the interval 7
6 6, .
20. Graph the function 22csc 1y x on the interval 3
2 2, .
2
3
2
2 5
2
3
21. Graph the function 1
sec 2 22
y x on the interval 2,4 .
22. Graph the function tan2 4
xy
on the interval 5
2 2, .
4
8
16
3
16
23. Graph the function 3cot 4 2y x on the interval 4,0 .
24. Graph the function 34tan 2y x on the interval 3,6 .
25. Determine the range of the following functions:
a) 3sin 2 7y x b) 2sec 2 11 8y x
26. Verify the identity 4 4 2cos sin 2cos 1x x x
27. Verify the identity 2 2
2 2
tan sec sec
cos sin sec 2cos
x x x
x x x x
.
28. Show that the equation cos2 cos sinx x x is not an identity by demonstrating that for a
specific value of x it is false.
29. Show that the equation sin2 sin cos 1x x x 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 true.
32. Verify the identity 2 2cos 90 sin sin 1 cosx x x x .
33. Find the exact value of cos 14 cos 29 sin 14 sin 29 .
34. Find the exact value of 512 4
512 4
tan tan
1 tan tan
.
35. Find the exact value of sin165 .
36. Verify the identity 2 tan tan
tan tan1 tan tan
x yx y y x
x y
.
37. Verify the identity
sin cot cot
cos 1 cot cot
x y x y
x y x y
.
38. Find the exact value of 2 2
12 12cos sin .
39. Find the exact value of 4sin22.5 cos22.5 .
40. Verify the identity 1 cos2
cotsin 2
xx
x
.
41. Verify the identity 2
2
1 tancos2
1 tan
xx
x
.
42. Find the exact value of 2
cot , if 5
tan2
and 90 180 .
43. Verify the identity 2
tan csc cotx x x .
44. Verify the identity 2
2
2
2
1 tancos
1 tan
x
xx
.
45. Find the exact value of 1 2sin
2
.
46. Find the exact value of 1 2sec
3
47. Find the exact value of 1 1tan 2cos
4
.
48. Find the exact value of 1 13 5cos sin cos
5 13
.
49. Find the exact value of 1 1sin 2sin
3
.
50. Solve the equation 2cos cos 2 0 on the interval 0,2 .
51. Solve the equation 24sin 1 0 on the interval 0,2 .
52. Solve the equation 2sec tan 1 on the interval 0,2 .
53. Solve the equation 2cos sin 2 on the interval 0,2 .
54. Solve the equation 2cos 0x on the interval 0,2 .
55. Solve the equation 2cos2 sin 0x x on the interval 0,2 .
56. Solve the equation 2tan sinx x on the interval 0,2 .
57. Solve the equation 1
sin cos2
x x on the interval 0,2 .
Sketch the solutions of the following polar coordinate equations.
58. 1 sinr 59. 1 2cos2r 60. 1 cos2r
61. 1 2cosr 62. cos4r
Find the points of intersection of the solution curves of the following pairs of polar coordinate
equations.
63. 1 cos , cosr r
64. 2cos3 , 1r r
Find the points of intersection of the curves defined by the following parametric equations.
65.
2
1
; 3 2
1
x t
t
y t
and
1
; 3 2
2 2
x s
s
y s
66.
2cos
;0 2
3sin
x t
t
y t
and 3 3
sec
;
tan
x s
s
y s
67.
cos
;0 2
sin 2
x t
t
y t
and
12
cos
;0 2
sin
x s
s
y s
68. Find the exact value of each part labeled with a variable.
69. 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?
30 60 x
y
z
w 8
70. If tan .75x and cos .8x , then find the value of tan cosx x .
71. Find the exact value of cos12
.
{Hint: 12 3 4
and cos cos cos cos cosA B A B A B .}
72. Find the exact value of 5
tan12
.
{Hint: 5
12 6 4
and
tan tantan
1 tan tan
A BA B
A B
.}
73. Find the exact value of 11
cos12
.
{Hint:1 cos
cos2 2
A A and
1111 6
12 2
.}
Find the exact value of the following:
74. 1 1sin sin
12
75. 1 4sin sin
3
76. 1 2cos sin
3
77. 1sin tan 2
78. 1 1tan cos
4
For each of the following, find sin x y , cos x y , tan x y , and the quadrant of x y .
79. 1 4
sin , cos10 5
x y , x in quadrant I, y in quadrant IV
80. 2 1
sin , cos3 5
y x , x in quadrant II, y in quadrant III
Find the sine and cosine of the following
81. B , given 1
cos28
B , B in quadrant IV 82. 2y , given 5
sec3
y , sin 0y
Find the following:
83. sin2
A
, given 3
cos4
A , with 90 180A b) sin 2x , given sin .6x , with
2x
84. sin y , given 1
cos23
y , with 2
y
Exactly solve the following trigonometric equations on the interval 0,2 .
85. 2sin 1x 86. 23cos 2cos 1 0x x 87. 4sec 2 4x 88. csc sin3 3
x x
89. sin sin2x x 90. cos2 cos 0x x 91. 2sin2 2cosx x 92. 2sin3 1 0x
93. cos 12
x 94.
6 4sin 2 cos 1x x 95. 26sin 17sin 12 0x x
96. Sketch the graph of the solution to the polar coordinate equation sin2r .
4
2
34
54
32
74
2
1
1
r
97. Sketch the graph of the solution to the polar coordinate equation 1 cosr .
98. Find the points of intersection of the solution curves of the polar coordinate equations
2 cos2r and 2 sinr .
99. Find the points of intersection of the solution curves of the polar coordinate equations
2sinr and sin cosr .
2 3
2 2
2
r
1
100. Graph the function tan 1y x on the interval 2 2, .
101. Graph the function sin 2y x on the interval 0, .
102. Determine the range of the function 8sin 5 7y x .
103. If 13
cos x , then find the exact value of sin tan sin cotx x x x .
Find the exact value of the following.
104. 1 4sin 2cos
5
{Hint: sin2 2sin cosA A A .}
105. 1 11 2sin sin sin
4 3
{Hint: sin sin cos cos sinA B A B A B .}
106. 112
1tan sin
3
{Hint: sin 1 cos
tan2 1 cos sin
A A A
A A
.}
107. 11 12 4
cos sin
108. Sketch the graph of the solution to the polar coordinate equation cos2r .
4
2
34 5
4 3
2 7
4
2
1
1
r
109. Sketch the graph of the solution to the polar coordinate equation 1 2sinr .
110. Find the points of intersection of the solution curves of the polar coordinate equations
1 sinr and 3sinr .
2
32
2
3
r
1
1
76 11
6
111. Find the points of intersection of the solution curves of the polar coordinate equations
2sin2r and 1r .
112. Find the area of the region that is inside the solution curve of 2sinr but outside the
solution curve of sinr .
113. Given that 4 3a i j and 2b i j and another vector 6 7r i j , find numbers k and
m so that r ka mb .
114. Express c in terms of a and b , given that the tip of c bisects the line segment.
115. For what values of x are 11xi j and 2xi xj orthogonal?
b
c
a
116. Given that a i xj and 2b i yj , find all values of x and y so that a b and a b .
117. Use the dot-product to show that an angle inscribed in a semi-circle is a right angle.
(Look at a b a b .)
118. 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.
Expand 2 2
a b a b by using the dot-product.
119. 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 0a b a b .
a b
a b a
b b
a
a
b b
a b
a b
a
b
b
a b
a b
120. 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 .
121. Find all vectors v in the plane so that 1v and 1v i .
Graph each parabola.
122. 2 4x y 123. 2
1 4 2y x
124. 2 8 8x x y
Graph each ellipse.
125. 2 2
125 16
x y 126.
2 24 161
81 25
x y 127. 2 26 5 30x y
128. 2 29 18 4 8 23 0x x y y .
Graph each hyperbola.
129. 2 2
116 25
x y 130.
2 24 1y x 131. 2 24 25 100x y
132. 2 29 18 4 8 31 0x x y y .
133. Find an equation for the parabola with focus of 4,4 and directrix of 2y .
134. Find an equation of the hyperbola satisfying the given conditions:
Endpoints of transverse axis: 4,0 , 4,0 ; asymptote 2y x
135. Solve the system
2 2
2 2
9
9
x y
x y
.
Solve the following systems of equations. Check to see if your answer agrees with the graph.
136. 2
1
1
x y (line)
y x (parabola)
137.
2 2 5
3 5
x y (circle)
x y (line)
138.
2 2
2 2
4 4
4 4
x y (hyperbola)
x y (ellipse)
139.
2 2
2 2
3 4 16
2 3 5
x y (ellipse)
x y (hyperbola)
140.
2
2
2 1
1
y x x (parabola)
y x (parabola)
141.
2
2 2
2
4 16
y x (parabola)
x y (ellipse)
142. Find the values of x and y in the figure.
143. Express the product of the following complex numbers in standard form.
a) 3 cos100 sin100 , 4 cos260 sin260z i w i
b) 2 cos20 sin20 , 6 cos25 sin25z i w i
144. Express the following in standard form.
a) 3
3 cos80 sin80i
b) 4
5 516 16
2 cos sini
145. On a recent episode of Who Wants to Be a Millionaire with Cedric the Entertainer, the
following question appeared.
For which of the following times will the minute and hour hands of a clock form a right
angle?
a) 4:05 b) 5:20
c) 3:35 d) 11:50
The contestant chose answer a) and he was told that he was correct. He wasn’t correct, in
fact, none of the options are correct. Let’s use basic trigonometry to find all the times for
which the minute and hour hands form a right angle. For t measured in minutes after
midnight, 30M t t represents the cumulative angle of the minute hand, and 360
H t t
represents the cumulative angle of the hour hand. In order for the two hands to form a
right angle, the difference between the cumulative angle of the minute hand and the
cumulative angle of the hour hand must be an odd multiple of 2 . So we get that
x
y 9
10 17
2
30 360 2
11360 2
2 1 ; 1,2,
2 1 ; 1,2,
2 1 ; 1,2,
180 2 1; 1,2,
11
M t H t n n
t t n n
t n n
nt n
a) Use the previous formula to find the number of times from one midnight to the next that
the minute and hour hands form a right angle.
{Hint: 180 2 1
# of minutes in a 24 hour period11
n .}
b) Use the same reasoning to find a formula for the times(in minutes after midnight) from
one midnight to the next(inclusive) that the minute and hour hands point in exactly the
same direction, and the number of times that it occurs.
c) Use the same reasoning to find a formula for the times(in minutes after midnight) from
one midnight to the next that the minute and hour hands point in exactly opposite
directions, and the number of times that it occurs.
146. a) Use geometry to fill in all the missing angles and sides in the following diagram of right
triangle ABC inscribed inside rectangle ADEF.
b) Use the diagram to find the exact values of the sine, cosine, tangent, cotangent, secant,
and cosecant of the angles 15 and 75 .
A F
C
E B D
30
45
60
45
90
90 90
90
3
2
1