Module 8: Practice Problems
1
Convert polar (5, π/3) to rectangular.
Solution
x=5cos(π/3)=5/2, y=5sin(π/3)=5√3/2. (5/2, 5√3/2).
2
Convert rectangular (0, −4) to polar.
Solution
r=4, θ=3π/2. (4, 3π/2).
3
Convert r=6sinθ to rectangular.
Solution
r²=6rsinθ ⇒ x²+y²=6y ⇒ x²+(y−3)²=9.
4
How many petals does r=3cos(5θ) have?
Solution
n=5 (odd), so 5 petals.
5
What type of curve is r=2+3cosθ?
Solution
Limacon with inner loop (b=3 > a=2).
6
Write z=−1+i in trig form.
Solution
r=√2, θ=3π/4. z=√2(cos(3π/4)+isin(3π/4)).
7
Find [2(cos30+isin30)]&sup4; using DeMoivre's.
Solution
2&sup4;(cos120+isin120)=16(−1/2+i√3/2)=−8+8i√3.
8
Find the square roots of 4(cos60+isin60).
Solution
w₀=2(cos30+isin30)=√3+i. w₁=2(cos210+isin210)=−√3−i.
9
Multiply z₁=3(cos45+isin45) and z₂=2(cos90+isin90).
Solution
6(cos135+isin135)=6(−√2/2+i√2/2)=−3√2+3i√2.
10
Give another polar representation of (3, π/4) with negative r.
Solution
(−3, 5π/4).