Express 5-2i in polar form. Express your answer in radians to the nearest hundreth.
r=sqrt5^2+-2i^2 which equals sqrt21.Arctan of -2/5+pi equals -18.7
Polar form=sqrt21 cis -18.7
Is this correct??
not quite
if x+iy = r e^iT = r (cos T i sin T)
then
r = sqrt (x^2+y^2)
not sqrt [x^2 + (iy)^2]
so r = sqrt (29), not 21
To express a complex number in polar form, we need to find its magnitude (r) and argument (θ). Let's calculate them step by step:
1. Magnitude (r):
The magnitude of a complex number z = a + bi, where a and b are real numbers, is given by the formula:
|r| = √(a^2 + b^2)
In this case, a = 5 and b = -2. Hence, the magnitude of 5 - 2i is:
|r| = √(5^2 + (-2)^2) = √(25 + 4) = √29.
So, the magnitude (r) is √29.
2. Argument (θ):
The argument of a complex number z = a + bi, where a and b are real numbers, is given by the formula:
θ = arctan(b/a)
In this case, a = 5 and b = -2. Hence, the argument (θ) is:
θ = arctan((-2)/5)
Using a calculator to find the arctan of -2/5, we get: θ ≈ -0.38050637711 radians.
So, the argument (θ) is approximately -0.3805 radians.
Now, we can express 5 - 2i in polar form:
Polar form = r cis θ
Polar form = √29 cis (-0.3805)
Therefore, the correct expression of 5 - 2i in polar form, rounded to the nearest hundredth, is √29 cis (-0.38) radians.