Given the rectangular-form point (–1, 4), which of the following is an approximate primary
representation in polar form?
A. −(4.12, 1.82)
B. (−4.12, −1.33)
C. (4.12, 1.82)
D. (4.12, 4.96)
Change 8 cis 240degrees to rectangular form.
A. -4(Square root 3)-4i
B. -8(Square root 3)-8i
C. -4-4(Square root 3i)
D. -8-8(Square root 3i)
(-1,4) is in quadrant II
tan^-1 Ø = -4
Ø = 104° or 1.82 radians
the length of r is √(1+16) = √17 = appr 4.12
mmmmhhhh? Can you do something with that?
8 cis 240°
= 8(cos240° + isin240°)
= 8(-1/2 + i(-√3/2)
= -4 - 4√3 i
To convert rectangular coordinates to polar coordinates, we can use the following formulas:
Magnitude (r):
r = sqrt(x^2 + y^2)
Angle (theta):
theta = arctan(y / x)
Let's apply these formulas to the given problem:
1. For the rectangular-form point (-1, 4), the polar coordinates will have a magnitude (r) and an angle (theta).
Magnitude (r):
r = sqrt((-1)^2 + 4^2)
r = sqrt(1 + 16)
r = sqrt(17)
Angle (theta):
theta = arctan(4 / -1) = arctan(-4)
theta ≈ -75.96 degrees (rounded to two decimal places)
The approximate polar form will be (r, theta) ≈ (sqrt(17), -75.96 degrees).
Now let's apply this process to the second problem:
2. To convert 8 cis 240 degrees to rectangular form, we can use Euler's formula:
x = r * cos(theta)
y = r * sin(theta)
Given: r = 8, theta = 240 degrees
x = 8 * cos(240 degrees)
x = 8 * (-0.5)
x = -4
y = 8 * sin(240 degrees)
y = 8 * (-sqrt(3) / 2)
y = -4 * sqrt(3)
The rectangular form will be -4 - 4 * sqrt(3)i.
Therefore, the answer to the second problem is option C (-4 - 4 * sqrt(3)i).