Practice Final Exam - Cumulative
Name:
Instructions:
Time yourself and aim for two hours.
Justify each answer.
S = rθ,
1
A = ab sin C
2
ν = rω,
sin(u ± v) = sin u cos v ± cos u sin v
cos(u ± v) = cos u cos v ∓ sin u sin v
tan u ± tan v
tan(u ± v) =
1 ∓ tan u tan v
sin 2u = 2 sin u cos u
cos 2u = cos2 u − sin2 u
= 2 cos2 u − 1
= 1 − 2 sin2 u
2 tan u
tan 2u =
1 − tan2 u
b2 + c2 − a2
cos(A) =
2bc
2
a + c2 − b2
cos(B) =
2ac
2
a + b 2 − c2
cos(C) =
2ab
a2 = b2 + c2 − 2bc · cos(A)
b2 = a2 + c2 − 2ac · cos(B)
c2 = a2 + b2 − 2ab · cos(C)
cos θ =
~u · ~v
||~u|| ||~v ||
z = r(cos θ + i sin θ)
z n = rn (cos(nθ) + i sin(nθ))
8π
1. Suppose θ = − . Find two coterminal angles in radians, one positive
3
and one negative.
2. Suppose θ =
5π
. Convert the angle to degrees.
9
3. Suppose θ = 330◦ . What number is the reference angle in degrees?
4. Sketch the angle θ = 420◦ .
For #5, #6, #7, #8, #9, and #10, use the figure to find the exact values
of the six trigonometric functions of the angle θ.
(6, 2)
θ
5. sin(θ) =
6. cos(θ) =
7. tan(θ) =
8. csc(θ) =
9. sec(θ) =
10. cot(θ) =
11. What is the exact value of
2
csc arccos
?
7
For #12 and #13, suppose a disc spins 20 revolutions per minute. A red
dot is 6 cm from the center. A blue dot is 2 cm from the center.
12. Give the angular speed in radians per minute of each dot.
13. Give the linear speed in cm per minute of each dot.
14. Give the amplitude, period, and phase shift of the function
f (x) = −3 cos(2πx − 3π).
15. Sketch a graph of the functions sin, cos, tan, csc, sec, cot.
sin(θ)
cos(θ)
tan(θ)
1
−2π
−π
1
π
2π
−2π
−π
−1
π
sec(θ)
−π
π
2π
−2π
−π
π
2π
−2π
−π
−1
16. Sketch a graph of the functions arcsin, arccos, arctan.
arcsin(x)
arccos(x)
arctan(x)
π
π
π
π
2
π
2
π
2
−1/2
2π
π
2π
1
−1
−1
π
cot(θ)
1
−π
−2π
−1
csc(θ)
−2π
2π
1/2
1
−1
−1/2
1/2
1
- π2
- π2
- π2
-π
-π
-π
17. Verify the identity
cos x sec2 x + sec2 x tan2 x = sec3 x
18. Find all solutions of the equation in the interval [0, 2π).
4 cos2 (x) − 3 = 0
19. Use the figure to find the exact value of sin(2θ) and cos(2θ).
(6, 2)
θ
20. A triangle has angle a = 3, b = 5, and C = 120◦ . Determine the exact
length of c and the exact measure of A.
(Side c should be simplified. Angle A may be written as arcsin(x) where
x is some number.)
21. Find the angle between ~u = h1, 3i and w
~ = h−3, −5i.
(The angle may be written as arccos(x) where x is some number.)
For #22, #23, and #24, suppose ~u = h−8, 2i and w
~ = h4, 3i
22. Find −2~u + 2w
~ in component form.
23. Find ~u − w
~ in component form.
24. Find the magnitude of ~u − w.
~
25. Write z = −6 + 6i in trigonometric form.
For #26 and #27, suppose
z1 = 3(cos (19◦ ) + i sin (19◦ )), and
1
z2 = 12
(cos (52◦ ) + i sin (52◦ )).
26. Find the product z1 z2 in trigonometric form.
27. Find z14 in trigonometric form.
√
28. A point in rectangular coordinates is given: (1, − 3)
Convert the point to polar coordinates.
29. A point in polar coordinates is given: (4, −7π/4)
Convert the point to rectangular coordinates.