Trigonometry Review
Angles in Degrees and Radians.
1. Convert θ = 140◦ to radians.
2. Convert θ = −10◦ to radians.
Name:
Arclength and Sector Area.
11. Find the arclength S and the area A of the
wedge. Give both an exact answer and a
calculator approximation
5. Use your calculator to convert 4 radians to
degrees. What quadrant is this angle in?
6. Find the smallest positive angle which is
coterminal with θ = 1100◦ .
7. Find the smallest positive angle (in radians)
which is coterminal with θ = −27π/4.
8. Find θ1 and θ2 in radians, assuming both
angles are between 0 and 2π.
S
r=
4. Convert θ = 7π/5 to degrees.
4c
m
3. Convert θ = 17π/12 to degrees.
A
◦
55
12. Suppose a wedge shaped area (like the
above) has A = 100 in2 and S = 10 in. Find
θ and r.
13. A large pizza has a diameter of 12 inches
and is cut into eight slices of equal size.
What is the area of one of the slices?
Trig Functions on Right Triangles.
14. Find cos θ, sin θ and tan θ for the triangle
shown:
θ1
θ2
2
θ
9. Find θ1 and θ2 in radians, assuming both
angles are between 0 and 2π (Hint: The
tick mark scheme is different).
5
15. Find sec θ, csc θ and cot θ for the triangle
shown:
θ2
1
θ1
θ
2
10. What quadrant are the following angles
in?
a. 11π/12
16. If θ is an acute angle such that sin θ = 3/4,
what is cos θ?
b. −300◦
17. If θ is an acute angle such that tan θ = 5/2,
what is sec θ?
d. 2.5 radians
18. If θ is an acute angle such that csc θ = 3,
what is sec θ?
c. 8π/5
M112/David Lane/Fall 2007
Applications.
Trig Functions on the Circle.
19. A 30 ft tall tree has a shadow which is 46 ft
long. What is the angle of elevation of the
sun?
20. The top of a 10 ft ladder rests against the
top of a 7 foot high wall. What is the angle
between the ladder and the ground?
21. An isoceles triangle has sides of length 4,
4 and 2 inches. Find the interior angles of
the triangle.
22. 80 feet from the base of a radio tower, the
angle between the ground and the top of the
tower is 37◦ . Find the height of the radio
tower.
24. If sin θ > 0 and cos θ < 0, which quadrant
is θ in?
25. If sin θ < 0 and cot θ < 0, which quadrant
is θ in?
26. If sec θ < 0 and tan θ > 0, which quadrant
is θ in?
27. Suppose θ is an angle in standard position,
and the terminal side of θ passes through
the point (2, −5). What are the exact values for the six trig functions?
28. Suppose θ is an angle in quadrant III such
that sin θ = −3/8. What are the exact values for cos θ and sin θ?
29. How many solutions are there to the equation sin θ = −.2 on the interval [0, 2π)?
What quadrants do the solutions lie in?
h
37◦
30. How many solutions are there to the equation cot θ = 1.5 on the interval [0, 2π)?
What quadrants do the solutions lie in?
80 ft
23. A 40 ft radio antenna is located on the top
of a building. 260 feet from the building,
the angle between the bottom of the antenna and the top is 13◦ . Find the height
of the building.
40 ft
31. How many solutions are there to the equation cos θ = .8 on the interval [0, 2π)? What
quadrants do the solutions lie in?
32. A fly lands on a circular ring of radius
2 inches at point Q(2, 0), then crawls 4.5
inches around the ring to position P . Find
the x and y coordinates of P .
P
4.5 in
h
13◦
Q(2, 0)
260 ft
M112/David Lane/Fall 2007
33. Fill in the blank with either < or >
Exact Values and Reference Angles.
Solve for t, 0 ≤ t < 2π in Probs 35 to 42:
ψ
θ
35. sin t =
1
2
√
2
36. cos t = −
2
37. tan t = 1
38. cot t = 0
a. cos θ
cos ψ
b. sin θ
sin ψ
c. cos θ
sin θ
d. cot θ
cot ψ
34. Use the unit circle below to estimate numerical values for the following, then check
your answer on a calculator:
a. cos 15◦
b. sin −40◦
c. sin 13π/24
d. cos 105◦
e. cos 5π/8
M112/David Lane/Fall 2007
√
3
2
√
40. cot t = − 3
39. sin t = −
41. csc t = −2
√
42. sec t = 2
43. Suppose cos 20◦ = a and sin 20◦ = b .
a. What are the cos and sin of 160◦ ?
b. What other angles between 0◦ and 360◦
have the same reference angle?
44. Find a formula for the area of an equilateral triangle as a function of its side x.