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.
8π 6π
2π
8π
+ 2π = − +
=−
3
3
3
3
8π
8π 12π
4π
θ2 = − + 4π = − +
=
3
3
3
3
θ1 = −
2. Suppose θ =
5π
. Convert the angle to degrees.
9
5π 180◦
5 180◦
θ=
=
= 5(20◦ ) = 100◦
9 π
9 1
3. Suppose θ = 330◦ . What number is the reference angle in degrees?
In the fourth quadrant, 360◦ − 330◦ = 30◦ .
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) = 2/ 40
√
6. cos(θ) = 6/ 40
7. tan(θ) = 2/6
√
8. csc(θ) = 40/2
√
9. sec(θ) = 40/6
10. cot(θ) = 6/2
r2 = 62 + 22 = 36 + 4 = 40
√
√
r = 40 = 2 10
11. What is the exact value of
2
?
csc arccos
7
cos(θ) =
adjacent
hypotenuse
22 + b2 = 72 → 4 + b2 = 49. So then b2 = 45 and b =
hypotenuse
csc θ =
=
opposite
√
45
7
√
45.
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.
(20)(2π) = 40π radians per minute.
(20)(2π) = 40π radians per minute.
13. Give the linear speed in cm per minute of each dot.
(40π)(6cm) = 240π cm per minute.
(40π)(2cm) = 80π cm per minute.
14. Give the amplitude, period, and phase shift of the function
f (x) = −3 cos(2πx − 3π).
Amplitude is | − 3| = 3.
Period is (2π)/(2π) = 1.
Shift is 3π/2π = 3/2.
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
cos x sec2 x + sec2 x tan2 x = cos x(sec2 x)(1 + tan2 x)
= sec x(1 + tan2 x)
= sec x(sec2 x)
= sec3 x
18. Find all solutions of the equation in the interval [0, 2π).
4 cos2 (x) − 3 = 0
4 cos2 (x) − 3 = 0
4 cos2 (x) = 3
cos2 (x) = 3/4
√
cos(x) = ± 3/2
From the unit circle values, x = π/6, 5π/6, 7π/6, 11π/6.
19. Use the figure to find the exact value of sin(2θ) and cos(2θ).
√
Pythagorean theorem
gives
r
=
40.
√
√
Since sin θ = 2/ 40 and cos θ = 6/ 40,
√
√
sin(2θ) = 2(2/ 40)(6/ 40) = 24/40
√
√
cos(2θ) = (6/ 40)2 − (2/ 40)2 = (36/40) − (4/40) = 32/40
(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.)
c2 = 32 + 52 − 2(3)(5) cos(120◦ )
= 9 + 25 − 30(−1/2)
= 34 + 15 = 49
c=7
52 + 72 − 32
2(5)(7)
25 + 49 − 9
=
70
65
=
70
65
A = arccos
70
cos(A) =
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.)
~u · w
~ =√(1)(−3) + (3)(−5)
= −3 − 15 = −18.
√
2
3
||~u|| = p1 + 3 = 10. √
||w||
~ = (−3)2 + (−5)2 = 34
−18
θ = arccos √ √
10 34
For #22, #23, and #24, suppose ~u = h−8, 2i and w
~ = h4, 3i
22. Find −2~u + 2w
~ in component form.
−2~u + 2w
~ = h16, −4i + h8, 6i = h24, 2i
23. Find ~u − w
~ in component form.
~u − w
~ = h−8, 2i − h4, 3i = h−12, −1i
24. Find the magnitude of ~u − w.
~
~u − w
~ = h−8, 2i − h4, 3i = h−12, −1i
||~u − w||
~ =
p
√
√
(−12)2 + (−1)2 = 144 + 1 = 145
25. Write z = −6 + 6i in trigonometric form.
|z| =
p
√
√
(−6)2 + (6)2 = 36 + 36 = 72
z is in the second quadrant. tan θ = y/x = (6)/(−6) = −1 for
θ = −45◦ , which is in the fourth quadrant. The correct angle is
θ = 180◦ + (−45◦ ) = 135◦
z=
√
72(cos(135◦ ) + i sin(135◦ ))
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.
1
z1 z2 = (3)
(cos(19◦ + 52◦ ) + i sin(19◦ + 52◦ ))
12
1
= (cos(71◦ ) + i sin(71◦ ))
4
27. Find z14 in trigonometric form.
z14 = (3)4 (cos (4(19◦ )) + i sin (4(19◦ )))
= 34 (cos(76◦ ) + i sin(76◦ ))
= 81(cos(76◦ ) + i sin(76◦ ))
√
28. A point in rectangular coordinates is given: (1, − 3)
Convert the point to polar coordinates.
√
r2 = x2 + y 2 = (1)2 + (−! 3)2 = 1 + 3 = 4. So r = 2
√
− 3
π
Then θ = arctan
=−
1
3
(r, θ) = (2, −π/3) (Or equivalently, (2, 5π/3)).
29. A point in polar coordinates is given: (4, −7π/4)
Convert the point to rectangular coordinates.
Recall −7π/4 is coterminal
to π/4.
√
√
√
√
x = r cos(θ) = 4(√ 2/2)
=
2
2
and
y
=
r
sin(θ)
=
4(
2/2)
=
2
2
√
Then (x, y) = (2 2, 2 2).