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Quiz for Math 112 (CRN 35805), Week 8
Note: Show your steps! These are the important basis for grading.
Name
- True or false. (30 points)
a) sin(cos−^1 x) =
1 − x^2 if − 1 ≤ x ≤ 1. T F
b) If |z| = r, and the argument(angle) of z is θ, then z = r · (cos θ + i · sin θ). T F
c) If z 1 = r 1 (cos θ 1 + i · sin θ 1 ), z 1 = r 2 (cos θ 2 + i · sin θ 2 ), then z 1 · z 2 =
r 1 r 2 (cos(θ 1 + θ 2 ) + i · sin(θ 1 + θ 2 )).
T F
d) If z = cos
2 π
12
2 π
12
, then z^12 = 1. T F
e) If z = a + i · b (a, b are real numbers), then z · z = a^2 + b^2. T F
f) The domain of f (x) = tan−^1 x is all the real numbers. T F
- Find all the solutions to the following equation. (20 points) (Fact: sin−^1 (
π
6
2 sin 3x = 1
- Express 1 + i ·
3 in polar form, and find the exact value of (1 + i ·
3)^24 (25 points)
- Find all the solutions to the following equation. (25 points)
sin x + cos 2x = 1
- Bonus problem (10 points): If we have the polar form of two complex numbers z 1 and z 2 ,
say, z 1 = r 1 (cos θ 1 + i · sin θ 1 ), z 2 = r 2 (cos θ 2 + i · sin θ 2 ), then the absolute value (or modulus) of
z 1 · z 2 is just r 1 · r 2 , and the argument(angle) of z 1 · z 2 is just θ 1 + θ 2 , pretty simple, isn’t it?
Now, if z = r(cos θ + i · sin θ), and z^3 = 8(cos
π
2
π
2
), what can you say about z? (Or in
other words, what can you say about the absolute value r and the angle(argument) θ?) (Hint, how
to express |z^3 | in terms of r? How to express argument of z^3 in terms of θ(the argument of z)?)
