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Question: How do you write the complex number in standard form \[\dfrac{3}{2}(\cos 300+i\sin 300)\]?...

How do you write the complex number in standard form 32(cos⁡300+isin⁡300)\dfrac{3}{2}(\cos 300+i\sin 300)?

Explanation

Solution

We are given an expression of complex numbers using trigonometric functions. We have to write the given expression of complex numbers in the standard form. We will first find the values of the trigonometric expression, that is, of cos⁡300∘\cos {{300}^{\circ }} and sin⁡300∘\sin {{300}^{\circ }}. Then, we will substitute the values in the expression and rewrite in the standard form of a complex number.

Complete step by step answer:
According to the given question, we are given a complex number expression but in trigonometric form. And we have to write this expression in the standard form.
The given expression we have is,
32(cos⁡300+isin⁡300)\dfrac{3}{2}(\cos 300+i\sin 300)-----(1)
Here, we will first have to find the values of trigonometric function present in equation (1), we get,
cos⁡300∘\cos {{300}^{\circ }}
⇒cos⁡(300∘−360∘)=cos⁡(−60∘)\Rightarrow \cos ({{300}^{\circ }}-{{360}^{\circ }})=\cos (-{{60}^{\circ }})
We know that cosine function is an even function.
An even function is of the form, f(−x)=f(x)f(-x)=f(x).
⇒cos⁡60∘=12\Rightarrow \cos {{60}^{\circ }}=\dfrac{1}{2}-----(2)
Also, we know that, cosine function in the fourth quadrant has positive values.
Also, we have,
sin⁡300∘\sin {{300}^{\circ }}
⇒sin⁡(300∘−360∘)=sin⁡(−60∘)\Rightarrow \sin ({{300}^{\circ }}-{{360}^{\circ }})=\sin (-{{60}^{\circ }})
We know that, sine function is an odd function.
An odd function is of the form, f(−x)=−f(x)f(-x)=-f(x).
⇒−sin⁡60∘=−32\Rightarrow -\sin {{60}^{\circ }}=-\dfrac{\sqrt{3}}{2}-----(3)
Now, we will substitute the equations (2) and (3), in the equation (1), we get,
32(12+i(−32))\dfrac{3}{2}\left( \dfrac{1}{2}+i\left( -\dfrac{\sqrt{3}}{2} \right) \right)
Now, we will simplify it further, we will get,
⇒32(12−i32)\Rightarrow \dfrac{3}{2}\left( \dfrac{1}{2}-i\dfrac{\sqrt{3}}{2} \right)
Now, we will open up the brackets and multiply each of the terms and we will get,
⇒32(12)−32(i32)\Rightarrow \dfrac{3}{2}\left( \dfrac{1}{2} \right)-\dfrac{3}{2}\left( i\dfrac{\sqrt{3}}{2} \right)
Solving the expression, we have,
⇒34−334i\Rightarrow \dfrac{3}{4}-\dfrac{3\sqrt{3}}{4}i
Therefore, the expression of complex numbers in standard form is 34−334i\dfrac{3}{4}-\dfrac{3\sqrt{3}}{4}i.

Note: The values of the trigonometric function of certain angles should be calculated correctly. And also while substituting the values back and evaluating the expression, make sure that it is done step wise as the trigonometric functions are at times tricky to handle.