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??
To express a complex number in polar form, you need to determine its magnitude (r) and angle (θ). Let's go through the steps:
1. Find the magnitude (r):
The magnitude of the complex number z = 5 - 2i can be calculated using the formula:
|r| = √(a^2 + b^2), where a and b are the real and imaginary parts of the complex number, respectively.
In this case, a = 5 and b = -2:
|r| = √(5^2 + (-2)^2)
|r| = √(25 + 4)
|r| = √29
|r| ≈ 5.39 (rounded to two decimal places)
2. Find the angle (θ):
The angle (θ) can be determined using the formula:
θ = arctan(b/a), where b and a are the imaginary and real parts of the complex number respectively.
In this case, a = 5 and b = -2:
θ = arctan(-2/5)
θ ≈ -0.38050637711 rad (rounded to two decimal places)
(Note: If you get a negative angle, simply add 2π to it to obtain a positive angle, since angles in polar form are typically positive.)
Now we have the magnitude (r ≈ 5.39) and angle (θ ≈ -0.38 rad) of the complex number.
3. Express the number in polar form:
To express the complex number z = 5 - 2i in polar form, we use the notation z = r * cis(θ):
z = √29 * cis(-0.38)
z ≈ 5.39 * cis(-0.38) (rounding r to two decimal places)
So, the correct answer is approximately:
5 - 2i ≈ 5.39 * cis(-0.38)