Pre-Calculus
Web Quiz 6.4-6.6 Answers
1). Find the polar coordinates that correspond to the point with rectangular
coordinates (1,3) .
y
tan (θ ) =
x = 1 and y = 3
x
2
2
2
θ = 71.56  or 1.25 radians
x + y =r
1+ 9 = r2
± 10 = r
Answer with θ in radians:
10 ,1.25 or − 10 ,4.39
(
) (
)
Answer with θ in degrees:
10 ,71.56  or 10 ,251.56 
(
) (
)
( 10 ,−108.44 )

 π
2). Find all possible polar coordinates for the point with polar coordinates  3,  .
 3
 π

 3, + 2πk 
 3

Answer:
where k is an integer.
π


 − 3, + (2k + 1)π 
3


3). Find the maximum value of r for r = 3 − sin θ , then find the corresponding angles
where these maximum r-values occur.
r=4
3π
(4,
)
2
 π 
 π 
4). Write the complex number in standard form a + bi : 5 cos  + i sin   
 4 
 4
 π 
 π 
5 cos  + i sin   
 4 
 4
 2
2

5
+i

2
2


5
5
2i
2+
2
2
5). Convert the equation 5 x − 2 y = 10 into polar form.
Answer:
5 x − 2 y = 10
5r cos θ − 2r sin θ = 10
r (5 cos θ − 2 sin θ ) = 10
10
r=
5 cos θ − 2 sin θ
6). Plot the complex number 5 - 2i in the complex plane and find its trigonometric form
where 0 ≤ θ < 2π .
r = 5 2 + (−2) 2 = 29
 −2 
 = −.38 radians
 29 
Final answer for θ = 5.90 radians
Trig form = 29 cos 5.90 + i 29 sin 5.90
θ = sin −1 
7). Find the 6th roots of the complex number 1 – i.
r = 12 + (−1) 2 = 2
7π
radians
θ=
4
n
θ + 2πk
θ + 2πk 

+ i sin
r  cos

n 
n

7π 8πk 
7π 8πk

+
+


6
4
4
4
4 

2 cos
+ i sin


6
6




7π
7π 

Ans: k = 0 gives 12 2  cos
+ i sin

24
24 

23π
23π 

k = 2 gives 12 2  cos
+ i sin

24
24 

13π
13π 

k = 4 gives 12 2  cos
+ i sin

8
8 

5π
5π 

2  cos
+ i sin

8
8 

31π
31π 

k = 3 gives 12 2  cos
+ i sin

24 
24

47π
47π 

k = 5 gives 12 2  cos
+ i sin

24
24 

k = 1 gives
12
4π
4π

8). Find the cube roots of 3 cos
+ i sin
3
3


.

4π
4π


+ 2πk
+ 2πk 


Answer: = 3 3  cos 3
+ i sin 3


3
3




Where k = 0, 1, 2
So:
3
3
3
4π
4π 

+ i sin
z 1 = 3 3  cos
, k = 0
9
9 

10π
10π 

+ i sin
z 2 = 3 3  cos
, k = 1
9
9 

16π
16π 

+ i sin
z 3 = 3 3  cos
, k = 2
9
9 

9). Find the distance between (5,120  ) and (−2,50  ) .
x 2 = 5 2 + 2 2 − 2(5)(2) cos 110 
x ≈ 6.0
10). Find the product of z1 and z 2 if z1 = 7(cos(25) + i sin (25)) and
z 2 = 2(cos(130 ) + i sin (130 ))
z1 • z 2 = 7 • 2[cos(25 + 130 ) + i sin (25 + 130 )]
z1 • z 2 = 14[cos(155) + i sin (155)]
π
π

11). If the cube root of a number is 2 cos + i sin  , find the number.
6
6

Answer: The number will be the 3rd power of the cube root.
3
 
π
π 
3π
3π 

 2 cos + i sin   = 2 3  cos
+ i sin

6
6 
6
6 

 
π
π
π
π

= 8 cos + i sin  = 8 cos + 8i sin
2
2
2
2

= 8(0) + 8i (1)
= 8i
12). Analyze the equation r = 3 − 4 sin θ in terms of the following:
Domain: (− ∞, ∞ )
Range: [− 1,7]
Symmetry: y-axis
Continuity: yes
Boundedness: bounded
Maximum r-value: 7
Asymptotes: none
3
 π 
 π 
13). Find  cos  + i sin    using DeMoivre’s theorem. Answer should be in
 4 
 4
standard form.
  π
 π 
z 3 = 13 cos 3 •  + i sin  3 • 
4
4 

 
2
2
z3 = −
i
+
2
2
14). Find the trigonometric form of the number − 5 + 4i .
− 5 + 4i
r=
Answer:
(− 5)2 + 4 2
= 41
 4 
 = −38.7° + 180° = 141.3°
 −5
θ = tan −1 
z = 41(cos141.3° + i sin 141.3°)
15). Convert the polar equation r = −4 cos θ into rectangular form and describe the
graph.
r = −4 cos(θ )
r 2 = −4r cos(θ )
x 2 + y 2 = −4 x
x 2 + 4x + y 2 = 0
(x 2 + 4 x + 4) + y 2 = 4
( x + 2 )2 + y 2 = 4
The graph is a circle with center (-2, 0) and radius of 2.