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Mu Alpha Theta National Convention: Hawaii, 2005
Trigonometry Topic Test – Alpha Division
1.
 
Evaluate: tan  
 24 
A) 2  3
B)
6  3  2 2
C) 2 3  6  2 2  3
D)
4  2( 3  1)
4  2( 3  1)
E) NOTA


2.
If sec( )  2 , which of the following could  equal?

3


7
A)
B)
C)
D)
4
3
6
4

3.
Simplify: sin 6 (x)  3sin 2 (x)cos2 (x)  cos6 (x)


A) 0

4.

5.

E) NOTA

B) 1

3
C) sin( x)  cos(x) D) 1 sin 2 (2x)
4
Which of the following equations has cos(20) as a root?


A) y  81x 4 144 x 2  64
B) y  8x 3  6x 1
C) y  10x 3  9x 2  30x  27
D) No such equation exists
E) NOTA
E) NOTA
 

, ) is a point in space
represented by cylindrical coordinates, what is the Cartesian
3 6
representation of the point?
If (4,
A) (1,
3 , 2 3 ) B) (2 3 , 2,


) C) (2, 2 3 , ) D) ( 3 , 3, 2)
3
6
E) NOTA
6. If f (x)  3sin( 4x)  5 , A is the amplitude of f (x) , B is the period of f (x) , C is the





D
 B  A?
maximum value of f (x) , and D is the minimum value of f (x) , what is
C

A)
7.

29
4
B)



C) 7 
2 
11 

4
2
D)

13 

4 2
E) NOTA
Which of the following angle measures is coterminal to 3650?
A) 130

B) 230

C) 40
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
D) 50
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Trigonometry Topic Test – Alpha Division
8.
Which of the following expressions is equal to sin( 33)  sin(11) ?
A) 4 sin 4 (11)  3sin 2 (11)
C) sin( 44)
9.

B) sin 2 (22)  sin 2 (11)
3
2
D)
 sin (33)  2sin (22)  sin(11)
E) NOTA
Evaluate tan(i  ln 2) where i 
 1 and the domain of the tangent function has been
expanded to include complexnumbers.
A)

3i
5
B) 
5i
3
C)
i
2cos(i ln 2)
D) No solution
10. What is the maximum of the real function f (x)  4sin x  3cos x ?



4
A) 4
B) 7
C) 5
D) arctan  
3

11. Given sin(A)=
A) 
E) NOTA
E) NOTA
5
15
, cos(B)= , what is tan(A+B)?
13
17
21
220
B)
19
20
C)
171
140
D)
91
110
E) NOTA
12. If f (x)  sin x , g(x)  cos x , and h(x)  f (x) g(x) . Find h( 8) .


2
A)
2

1
B)
2


C)
2
D)

2
4
E) NOTA
13. Which of the following describes the behavior of the graph y  sin x at x   2? (where

integer function)?



 x  is the greatest
A) continuous
B) removable
discontinuity
C) jump 
discontinuity
D) infinite

discontinuity
E) NOTA

3 
14. What is the phase shift of y  sin 3x 
?

8 
A)

3
8
B)



8
C) 

D) 
8


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3
8
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Trigonometry Topic Test – Alpha Division
15. In ABC, BC  4 , mBAC  60 , and mABC  75 . What is the length of AB ?
A) 2 6


B)
4 6
3

C)
6 22 6
3
D) 4 2

16. What is the name of the polar graph of r  sin(  )  cos( ) ?




A) lemniscate
B) cissoid
C) limacon
D) conic
E) NOTA
E) NOTA

17. What is the value of cos( ) if  is the acute angle between the vectors <3,4> and <2,6>?
10
3
A)

10
B)
 25
C)
1
2
D)
3 10
10
E) NOTA
 
 
 
 
 
 
18. Evaluate: cos(0)  sin(0)  cos    sin    cos    sin    cos    sin  


 3 
6 
6
3
2
2
B) 2  2  3
A) 0
D) 3  3
C) 1
E) NOTA
19. If sin( x)  cos( x)  tan( x)  cos(x)  sec( x)  0 , what is the sum of all possible values for x
on the interval


 0,2 ?


A) 3

B) 5
C)
9
2
D) 0
E) NOTA
20. How many of the following are equivalent to cos(2x) ?



cot 2 (x) 1
ix
ix

 e e
 1 sin 2 (2x)
 cos4 (x)  sin 4 (x)
2
cot 2 (x)  1


A) 1

B) 2


C) 3

D) 4
E) NOTA

21. In the figure below A and B lie on the circumference of circle C, if AB  AC  BC  1, what
is the area of the shaded region (the region between the chord AB and the arc AB)?
A)


3

3
2
B)   3

C)

6

3
4

D)

Page 3 of 5

12

3
8
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Trigonometry Topic Test – Alpha Division
22. Evaluate sin( x) in the figure to the right.
A) .8
B) .75
C) .6

D) Not enough
information
23. Find the measure of the acute angle between the x-axis and the line r 
5
A) arctan  
2
2
B) arctan  
5
C) arctan 10
arcsin( 2)
D)
 arccos(5)
E) NOTA
1
.
2sin   5cos
E) NOTA
24. Two particles are at the same location at t = 0s and then begin to move away from each other

at an angle of 60. If one particle is moving at 4 m/s
 and the other at 3 m/s, what is the
square of the number of meters between them at t = 2s?
A) 10
B) 13
C) 52
D) 100
E) NOTA
D) 4
E) NOTA
25. What is the area of the graph enclosed by r  sin  ?
A)

4
B)

2
C) 

26. A right triangle has a hypotenuse of 4 and an angle with a sine of 1 3 . What is the length of
the longer leg

 of the triangle?
A)
4
3
B)
32
9
C)
8 2
3
D) 12
E) NOTA
27. Which of the following is equivalent to cis(12 )  cis(4)?

A) cis(3)

B) cis(8)

C) cis(16)

Page 4 of 5
D) cis(48)
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Trigonometry Topic Test – Alpha Division
28. Which of the following is true for all values of x for which both the left and right sides of
the equation are defined?
sec(x)  sin( x)  tan( x)
 tan( x)
I.
csc(x)  cos(x)  cot( x)
sin( 3x) cos(3x)

 4 cos(2x)
II.
sin( x)
cos(x)
cos(4 x)

 2cos(x)  (sin( x)  2sin 3 (x))
III.
cot(4 x)

A) I only
B) II only
C) III only
D) I, II, and III
E) NOTA

29. A rhombus has an area of 24 square units and one of its diagonals, d, has a length of 8 units.
If x is an angle not intersected by d, which of the following is equal to sin(x)?
A)
7
25
B)
4
5
C)
24
25
D)
8
5
E) NOTA
30. Order the following numbers from least to greatest: cos(90), tan(/3), cot(315), sin(3/4)



A) cot(315), cos(90), tan(/3), sin(3/4)
B) tan(/3), sin(3/4), cos(90), cot(315)
C) sin(3/4), cos(90), cot(315), tan(/3)
D) cot(315), sin(3/4), cos(90), tan(/3)
E) NOTA
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