Download Trigonometry Review Worksheet

AB Calculus - Hardtke
Competency Review 2: Trigonometry
Name __________________________________
*Hint: If you have trouble with any of these trig questions, use Ms. H's Interactive Slideshow Review of
Trigonometry, also available from our class website.
NO CALCULATOR for any problem on this worksheet unless the direction states otherwise.
1. Write at least four Pythagorean Triples – no two from the same "family" of similar triangles.
2. Use generic variable expressions to represent the relationship of the sides of all 30-60-90 triangles?
3. Use generic variable expressions to represent the relationship of the sides of all 45-45-90 triangles?
4. In every 30-60-90 ∆, what is the ratio of the long leg to the hypotenuse?
5. In every 30-60-90 ∆, what is the ratio of the short leg to the hypotenuse?
6. In every 45-45-90 ∆, what is the ratio of a leg to the hypotenuse?
7. In every 45-45-90 ∆, what is the ratio of the two legs?
8. Find the missing length in each triangle below.
c
15
30o
a
30
30o
o
45
o
45
14
15
b
14
d
f
o
45
2 10
o
e
15
9. What is the definition of radian?
10. Convert each degree measure to radians.
a. 30o
b. 45o
c. 60o
d. 90o
e. 135 o
f. 180o g. 270o
h. 300o
i. 360o
7
6
h. – 3 
i. 
j. 420 o
k. 960 o
11. Convert each radian measure to degrees.
a.
3
2
b. 
c. –
5
3
d.
5
4
e.

3
f.
5
6
g. 
12. Write two positive and two negative radian measures that are coterminal angles to

j.
2
11

6
k.
7

2

.
3
13. Write two versions of the definitions of each trig ratio in the chart below:
sin
cos
tan
sec
csc
cot
Right Triangle Def:
(use when hyp  1)
Unit Triangle Def:
(use Orly when hyp =1)
14. Use a calculator for this problem and round to three decimal places. Find the angle of elevation with the floor if a 15.5 foot wheelchair
access ramp must reach a height of 3 feet vertically.
15. Use a calculator for this problem and round to three decimal places. Find the height of a movie screen if when you are 30 feet
(horizontally) from the screen you must raise your eyes 42 o to see the bottom of the screen and 57o to see the top of the screen. (Hint: 2
trig problems and subtract?)
16. Find the EXACT value for each angle.
Angle in radians
sin x
cos x
tan x
sec x
csc x

6
3
x=
2
x=
x=

3
x= 
x= 

4
2
3
x = 3
x=
7
6
17. Graph each trig function. Be sure to label key coordinates on each axis.
A. y = sin x
B. y = cos x
C. y = tan x
D. y = csc x
E. y = sec x
F. y = cot x
 x
18. Graph y  3sin   below. Be sure to label all key points on the x and y axes.
2
Amplitude:______
Period:_________
cot x


19. Graph f ( x)  cos  x    1 below. Be sure to label all key points on the x and y axes.
2

Amplitude:______
Period:_________
20. Give the domain and range of y = cos x.
21. Give the domain and range of y = tan x.
 
22. Since the point is on  ,1 y = sin x, what related point is on y = arc sin x?
2 
23. Since the point  , 1 is on y = cos x, what related point is on y = cos-1x?
24. Graph each inverse trig function. Be sure to label key coordinates on each axis.
A. y = sin-1x
B. y = arc cos x
Hint: Recall that inverse functions are
reflections of each other across
the diagonal y = x since the
domain (x's) of one becomes the
range (y's) of the other. Only 1-1
functions have inverses, so we
select a section of the original
function that passes both the
vertical and horizontal line test.
Go to the review slideshow if you
need more help with inverses.
-1
C. y = tan x
25. Give the equations of two asymptotes on the graph of y = tan x and the equations of two asymptotes on the
graph of y = tan-1x.
26. Solve each equation over [0, 2 ):
x 1
A. cos 
B. 2sin(3x) –
2 2
3 =0
C. 3tan 2 x  1
27. Find ALL solutions of x given tan x < 0 and 4cos2x – 1 = 0.
28. Simplify each expression over the proper domain and range of the function.

1
2
A. arc cos  
B. sin 1
C. tan 1  1

2
 2 
D. sec1 2
29. Write a PYTHAGOREAN IDENTITY involving each ratio shown below.
A. sin
B. csc
C. tan
30. Complete each COFUNCTION IDENTITY below.


A. cos   x  
2



B. sec   x  
2



C. cot   x 
2

31. (Use results from #30.) If sin
7
a
1
  , find cos  .
18
b
9
32. Use a NEGATIVE ANGLE IDENTITY to answer each true/false question below.
T
F A. For all values of x, cos x = cos (– x). Hint: Think of cos
T
F B. For all values of x, sin x = sin (– x).
T
F C. For all values of x, tan x = tan (– x).
2
2
and cos (–
), for example.
3
3
33. Complete the DOUBLE ANGLE IDENTITY for each.
A. sin 2x =
B. cos 2x =
C. tan 2x =
True or False.
T
F 34. The graph of y = – tan x is a reflection of y = tan x across the y-axis.
T
F 35. In quadrant II cosine is positive.
T
F 36. In quadrant IV tangent is positive.
T
F 37. If the lengths of the legs of a right triangle are 5 and 12, then the hypotenuse is 13.
T
F 38. The point on the unit circle that corresponds to
T
F 39.
T

5
and 
are coterminal angles.
3
3


F 40. and – have the same cosine value.
T
4
4
F 41. The amplitude of y = 2 sin 3x is three.
T
F 42. The period of y = 3 sin 4x is 2  .
T
F 43. (0, 1) is a point on the graph of y = cos x.
T
F 44. y = cot x has a vertical asymptote at x = -1.
T
F 45. For all values of x, 1 + cot2 x = csc2x.
7
radians is
6

3 1
,  
 
2
 2