Consider w = 2(cos(210°) + i sin(210°)) and z = 2(cos(330°) + i sin(330°)). What is w – z expressed in rectangular form?
Answer (2)
The difference w - z in rectangular form is -2√3. First, both w and z were converted from polar form to rectangular form. The final calculation shows that the result is a real number with no imaginary component.
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To solve for w − z expressed in rectangular form, we'll start by examining the given complex numbers w = 2 ( cos ( 21 0 ∘ ) + i sin ( 21 0 ∘ )) and z = 2 ( cos ( 33 0 ∘ ) + i sin ( 33 0 ∘ )) , which are expressed in polar form.
First, let's convert each complex number into rectangular form:
For w :
cos ( 21 0 ∘ ) = − 2 3
sin ( 21 0 ∘ ) = − 2 1
So, w = 2 ( − 2 3 + i ( − 2 1 )) = − 3 − i .
For z :
cos ( 33 0 ∘ ) = 2 3
sin ( 33 0 ∘ ) = − 2 1
So, z = 2 ( 2 3 + i ( − 2 1 )) = 3 − i .
Next, we calculate w − z :
( w − z ) = ( − 3 − i ) − ( 3 − i ) = − 3 − i − 3 + i
Simplifying further:
The real parts are: − 3 − 3 = − 2 3
The imaginary parts are: − i + i = 0
Therefore, w − z in rectangular form is:
− 2 3
