Question

How do you express the complex number in rectangular form:$z = 5\left( {\cos {{120}^ \circ } + i\sin {{120}^ \circ }} \right)$?

Answer 1

Hint: The above question is based on the concept of trigonometric form of complex numbers. The main approach towards solving this question is to convert the polar form into the rectangular form by knowing the point values of the cosine and sine function in the equation and substituting it we get the rectangular form.

Complete step by step answer:
The above given equation is in polar form. The polar form of a complex number is a way of expressing or representing a complex number. The polar form generally can be written as
$z = r\left( {\cos \theta + i\sin \theta } \right)$
where r is the absolute value(hypotenuse) and it contains trigonometric functions like sine and cosine.This polar form is derived from the rectangular form which can be written as:
\[z = a + ib\]
Here a is the adjacent side in a triangle and b is the opposite side of the angle.
So now we need to convert polar form into rectangular form.
Now by using the formula to get the coordinates a and b,
\[a = r\cos \theta \\
\Rightarrow b = r\sin \theta \]
Here the value of r is 5 and the angle is 120.Therefore by substituting it we get,
\[a = 5\cos {120^ \circ } = 5 \times \dfrac{{ - 1}}{2} = \dfrac{{ - 5}}{2} \\
\Rightarrow b = 5\sin {120^ \circ } = 5 \times \dfrac{{\sqrt 3 }}{2} = \dfrac{{5\sqrt 3 }}{2} \\ \]
So, the rectangular form can be written as given,
\[\therefore z = \dfrac{{ - 5}}{2} + i\dfrac{{5\sqrt 3 }}{2}\]

Note: An important thing to note is that the polar form is derived from the rectangular form in such a way that the basic trigonometric ratios are applied in the following way where \[\cos \theta = \dfrac{a}{r}\] and \[\sin \theta = \dfrac{b}{r}\]in which when r taken on the left hand side gives the formula to convert into rectangular form.