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Trigonometry Practice Problems
No calculator unless specified otherwise.
1. How many sides does a triangle have?
3
2. What is the sum of the 3 angles in a triangle?
180
3. Define the following types of triangles:
a. Equilateral
All sides same
b. Isosceles
2 sides same
c. Scalene
3 sides different
4. What is the area of a triangle?
5. 210
a. Sketch this angle in standard position.
b. What is the reference angle?
6. Convert 120 to radians.
7. Convert
5
to degrees.
3
8. The minute hand of a clock completes one revolution in an hour.
How many radians does the minute hand move in 120 minutes?
9. The minute hand of an analogue clock completes one revolution in 1 hour.
Determine the exact value of the angle, in radians, the minute the hand
moves in 135 min.
9 /2
10. What is the first positive co-terminal angle to
11. True or False:
/ 3 is co-terminal to
45 in radians?
11
3
12. Radians vs. Degrees
a. Define radian
b. When to use radians vs. degrees?
c. 1 radian is approximately how many degrees?
13. Convert
14. If no angle unit is specified, what unit is assumed?
Radians
15. Rationalize:
a.
b.
1
2
1
3
2
2
3
3
16. Special angles:
a. Sketch the 30-45-60 triangle and explain where it comes from.
b. Sketch the 45-45-90 triangle and explain where it comes from.
17. Evaluate:
a.
sin 30
1/2
b. sin 45
c.
2 /2
3/3
sin 60
18. Evaluate:
a.
cos
b. cos
c.
cos
6
4
3
19. Evaluate:
a. tan 30
b. tan 45
c. tan 60
20. Evaluate:
a.
sin 120
3
b. cos
4
c. tan 210
d. sin 0
e. cos
f.
g.
tan
2
sin( 810 )
21. Use www.desmos.com to reason why the identity tan
22. f (x )
sin
co s
is true.
sin
a. Sketch the overall shape of this function beyond
360
360
b. Describe one application of understanding this function.
c. Why can’t you see the overall shape of y
d. Sketch f (x ), 0
x
sin
in the default calculator degree mode?
720
e. Label quadrants and explain why ASTC works.
f.
23. y
The independent variable
is often t where t represents what?
cos x
a. Sketch cos x, 0
x
2
b. Write cos x as a transformation of sin x and vice-versa.
24. Graph f (x )
3 sin x and g (x )
a. Amplitude?
b. Period?
c. Domain?
d. Range?
0.5 sin x on the same graph.
e. For what values is g (x )
25. y
f (x ) ?
sin 4x
a. What is the b-value?
b. What does b represent?
c. What is the period?
d. Sketch
26. f (x )
2
cos
3
a. Period?
b. Sketch over two cycles
27. Where is tan
28. Sketch y
29. y
located on the unit circle?
tan ,
2
2
tan x
a. Domain?
x
n
2
b. Range?
y
c. First positive vertical asymptote?
d. General equation of vertical asymptote?
/2
x
2
n
e. Period? Compare with period of sine and cosine functions.
f.
Period formula?
g. Sketch two cycles of this function in degrees.
30. Sketch the reciprocal trigonometric functions:
a. y
csc x
i. Domain?
ii. Range?
per
b
b. y
sec x
c.
cot x
y
i. Domain?
ii. Range?
iii.
f (x )
tan x . Write g (x )
cot x as a transformation of f (x ) .
31. Evaluate:
a.
sin 2 (30 )
b. csc 3
c.
4
4
sec2
3
3
d. cot2
1
4
32. See diagram below:
a. Find a relationship between d and
b. What is the meaning of
c. Can
d. Given d
33. sin
.
d
0?
6000 tan
Plane is above
90 ?
no
12000 m , find
in degrees.
63.4
1
2
a. General solution?
b. Solve
within the domain
2
2
3
34. cos
. Find , 0
2 .
2
1 . What is the general solution for
35. sin
36. sec t
37. tan x
2 . Solve 0
t
?
4
3
a. General solution in radians in concise notation?
2
x
n or
3
x
3
n
b. Solve within the following domain: (
38. Suppose tan2
tan
0 and 0
,3 )
2 . What does
39. What is the general solution of the equation 2 cos
40. Solve the equation sec2
sec2
2
3
1
equal?
41. Sketch tan
2
2
3
0,
0 in degrees?
,
5
3
,
4
,
,
8
3
5
4
120
360n
and 240
360n
0,
0,
2
,
/ 4,
/ 4,
3
4
,
3
4
x in radians:
42. What is the equation of a unit circle?
43. What is the equation of a circle about the origin with a radius of 25?
44. (x
2)2
(y
3)2
9
a. Sketch
b. Domain?
45. y
9
(x
3)2
c. Sketch
d. Length?
46. On a unit circle y
sin , whereas x
?
47. On a circle with radius r, the (x , y ) coordinate is what?
x
cos
(r cos , r sin )
48. Derive the arc of circle formula arc
r
49. The arc of a circle is 5 m. The angle is 45 . What is the radius of the circle?
50. What is the exact coordinate of the point that lies at the intersection of the terminal arm and the
circle at an angle of 150 on the unit circle?
51. How many full rotations are completed if you rotate 400
52. Where does y
radians?
2x intersect the unit circle in Quadrant III?
53. The point ( 12 p, 5 p) is on the arm of an angle in standard position. Evaluate cot .
54. P ( )
1
2
3
,
2
3
55. If P ( )
2
,
. What is the coordinate of P (
1
2
, what are the coordinates of P (
56. Given that sin
0.7 , find sin(
57. Given that sin x
0.3 , find cos
)?
1
/ 2)
2
,
3
2
)
2
x
58. What point on the unit circle corresponds to cos
2
and is in Quadrant II?
2
59. The point (3a, 4a ) is on the terminal arm of an angle in standard position.
State the exact value of the six trigonometric ratios.
60. Determine tan
if sin
4
5
and cos
4 / 5 , cos
3 / 5 , tan
4/3
csc
5 / 4 , sec
5 / 3 , cot
3/4
0.
4
61. Given cos
sin
0 . 4/3
, find cot if tan
5
62. See diagram below to find the measure of the angle in standard position
(angle formed from the x-axis to the terminal arm):
63. Given
1
2
1
3
a
a
, evaluate
csc 3 ( / a )
a2
64. The following circle is centered on the origin. Point A is rotated
90 . What are the exact
coordinates of the new point?
1,
3 1
, . Find the coordinates of P (
2 2
65. P ( )
66.
3
) on the unit circle.
2
3
1
,
2
120 in standard position on a circle with radius 10.
Find the exact coordinate on the circle.
3
67. sin
2
5, 5 3
1
. What is the exact coordinates on the unit circle?
2
,
3
2
,
1
2
3
,
2
68. The terminal arm is in Quadrant I.
The x-value is
69. Given
1
70. If sin A
12 and the y -value is 2. Find
x
1
5
2
3
2
x
, evaluate sec2
1
and cos B
2
.
30
5x
4
30
, determine the exact value of sec A
csc B .
6
Write as a single fraction and rationalize the denominator.
1.
Sketch y
2.
Sketch f (x )
3.
What is the equation of the following graph?
3 sin 2 x
4
2 cos 2x
2 3
3
2.
. Amplitude?
y
4.
Suppose the water depth is given by the formula: d (t )
t 10 ,
6
where t is the time, in hours, after the first high tide. What is the period?
5.
See above. A cruise ship requires 12 m of water to dock safely. Determine
6 cos
the number of hours per cycle the ocean liner can safely dock.
2 sin 2x
2
6.
What is the period of tan x ?
7.
What is the general equation of the asymptotes of tan x ?
8.
Sketch tan x without a calculator.
9.
What is the period formula for sin x and tan x ?
10. What is the equivalent to
radians or 180
1
sin x
,
1
cos x
,
1
tan x
x
2
n or 90
P
2
b
?
11. Evaluate sin 2 (30 )
12. Evaluate cos 3
13. Evaluate tan
14. Evaluate csc
/4
6
3
15. Evaluate cot
4
16. What are the vertical asymptotes of f (x )
17. Sketch
tan x for
2
x
2 ?
2 sin(4 x ) for at least one full period.
18. 2 sin(2 )
2 . Find 0
2
cot
2
2 tan
19. Solve cot
20. Solve sec
2
0. State the solution in general form.
3
0 . State the general solution to the nearest degree.
21. What is the graph of the function below?
22. What is the period of tan(2x ) ?
23. What is the amplitide and period of y
24. The graph of y
3 sin(4 x ) ?
cos x can be obtained by translating the graph of y
25. Describe the transformation y
26. Ferris wheel height formula: h(t )
3 sin
3x
16 cos
t
60
3
2
20 .
What is the diameter of the Ferris wheel? h metres, t seconds.
27. In the Ferris wheel question above, what is the minimum and maximum
height? How fast are passengers moving?
sin x in what way?
, P
180n
b
28.
4 sin x
2
0 . Sovle the general solutions for x in degrees.
1
29. What are the solutions to sin 2 x
x 360 ?
0, 0
2
30. What is the exact value of sec x ? (rationalize the denominator if necessary)
31.
Circular Measure and Trigonometric Functions
Trigonometric Equations and Identities
1. What is csc
2.
What is tan x as a quotient identity?
3.
What is the solution for 2 cos x
4.
3
cos x
0 for 0
Show that the Pythagorean identity sin x
is equivalent to tan x
Prove cot x
2
x
2
cos x
2 ?
1,
Divide both sides by cos2 x
sec x .
1
sin
sin x
2
2
5.
1
as a reciprocal identity?
Divide both sides by cot x .
cot x sin x csc x
Rewrite csc x as
6.
Write two equivalent forms to cot x
7.
Show that
8.
Show that
9.
cos x
sin x
cos x
1
cot x
sin x cos x
Try breaking down: ex. cot x
tan x
Hint: tan 2
sin2 x sec2 x
10. Show that sin2 x tan2 x
tan2 x
sin2 x
11. Show that 2 sin x cos x
1
(cos x
12. Show that 2 cos2 x
1
2 sin2 x
1
13. What cos(a
b ) equivalent to?
14. Given tan(x
y)
tan x
tan y
A
sin x
cos x
tan x
sin x
(sin x 1)
Try mult. LS by
(sin x 1)
sin x
1
1
Show that tan2 x
1
1
sin x )2
sin 2 x
cos 2 x
, sec 2 x
Hint: tan 2 x
cos x
sin x
1
cos 2 x
sin 2 x
cos2 x
Try FOILing RS
Hint: sin 2 x
cos2 x
1
See formula sheet
, solve A .
See formula sheet
tan
tan
15 .
1 tan
tan
10
15
16. What is the exact value of cos15 ?
15. Find the exact value of
10
17. What is the exact value of cos 165 ?
18. Simplify cos 14 cos 172
tan(114 )
1
2
cos
9
tan( 24 )
2
sin
9
18
.
is undefined.
Hint: look at the tan(2A) formula.
2 sin 2 x
sin 4x
Know that tan 2x
cos 4x
1
23. Show that the solution to sin x
4 csc x
2
24. Show that the solutions for tan a
0 over the domain 0
5
2
is 3 / 2
3 over the domain
3 sec a
25. Show that the general solution, in radian, to sin 2
27. Show that
cos x sec x
sec x
B)
1
tan A tan B
cos(A
B)
1
tan A tan B
4 cos3 x
2
n
0
3 cos x
29. Show that the general solution to 2 cos a cos 2a
120n and 80
0 is
2 co s
tan x sin x
tan x
cos2 x tan 2 x
cos(A
28. Show that cos 3x
is 40
x
360 is 120 , 180 , and 240 .
26. Show that
2 sin a sin 2a
1
120n
30. Show that the general solution, in radians, to the equation
(4 cos2 2x
1) sin
31. Show that tan
60 )
sin(2x )
22. Show that tan 2x
x
Hint: cos(225
Use formula
sin
18
tan(114 ) tan( 24 )
21. Show that cot x
0
45 )
sin 14 sin 172
19. Find the exact value of cos
20. Show that
Hint: cos(60
1
3
0 is 3 n
x
1
2
cos
sin
32. Determine the value of cos
sin sec
tan
sec tan
33. Use a counterexample to show that cos(x
34. Prove the identity tan
35. Is the equation csc
2
2
sin
1
2
2
cot
36. Verify graphically that cos x tan x
sin
2
y)
tan
2
cos x
sin x for 0
85
x
and cos
84
85
.
cos y is not an identity.
.
true for all values of
Geometry of Triangles and Circles
13
if sin
?
360 .
sin 2x
cos 2x
.