math 2412 activity 4(due with final exam) 1. use ...nhmath.lonestar.edu/faculty/hortonp/math...

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

17. Graph the function 3cos 4 1y x on the interval 4 4, .

4

8

8

4

6

17

6

3

2

5

6

13

6

18. Graph the function 34 8

1sin 2

2y x on the interval 17

6 6, .

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 .


tan csc cotx x x .


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 .}


tan12

.

{Hint: 5

12 6 4

and

tan tantan

1 tan tan

A BA B

A B

.}


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 .


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 .


1 sinr and 3sinr .

2

32

2

3

r

1

1

76 11

6


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

math 2412 activity 4(due with final exam) 1. use ...nhmath.lonestar.edu/faculty/hortonp/math...

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