Question

How do you express the complex number in rectangular form: $12\left( {\cos 60^\circ + i\sin 60^\circ } \right)$?

Answer 1

Given a trigonometric expression. We have to write the expression in rectangular form. We will first determine the value of sine and cosine function for the particular measure of angle. Then, substitute the values and determine the rectangular form of the corresponding polar form.
Formula used:
The rectangular form of corresponding to the polar form $r\left( {\cos \theta + i\sin \theta } \right)$ is given by:
$a + ib$
Where $a = r\cos \theta$ and $b = r\sin \theta$

Complete step by step solution:
We are given the trigonometric expression in polar form, $12\left( {\cos 60^\circ + i\sin 60^\circ } \right)$
First, we will compare the equation with standard form of polar equation, $r\left( {\cos \theta + i\sin \theta } \right)$ to determine the value of $r$, $\cos \theta$ and $\sin \theta$.
$\Rightarrow r = 12$
$\Rightarrow \cos \theta = \cos 60^\circ$
$\Rightarrow \sin \theta = \sin 60^\circ$
Now, we will calculate the value of $a$ by substituting $r = 12$ into the relation $a = r\cos \theta$.
$\Rightarrow a = 12\cos 60^\circ$
Now, we will substitute $\dfrac{1}{2}$ for $\cos 60^\circ$.
$\Rightarrow a = 12 \times \dfrac{1}{2}$
$\Rightarrow a = 6$
Now, we will calculate the value of $b$ by substituting $r = 12$ into the relation $b = r\sin \theta$.
$\Rightarrow b = 12\sin 60^\circ$
Now, we will substitute $\dfrac{{\sqrt 3 }}{2}$ for $\sin 60^\circ$.
$\Rightarrow b = 12 \times \dfrac{{\sqrt 3 }}{2}$
$\Rightarrow b = 6\sqrt 3$
Now, we will write the values of a and b into the rectangular form of the equation, $a + ib$
$\Rightarrow 6 + i6\sqrt 3$
Hence, the complex number $12\left( {\cos 60^\circ + i\sin 60^\circ } \right)$in rectangular form is $6 + i6\sqrt 3$

Additional Information: The complex number can be written in two forms: polar form and rectangular form. The polar form can be represented as $r\left( {\cos \theta + i\sin \theta } \right)$ and the corresponding rectangular form is represented as $a + ib$ where $a = r\cos \theta$ and $b = r\sin \theta$ where r is the length of the vector and theta is the angle made by the vector with real axis.

Note: In such types of questions, the students mainly don't get an approach on how to solve it. In such types of questions students forgot to apply the relation between the rectangular form and polar form of the complex number using the basic trigonometric ratios.