Question

Which point is represented by 6 times the quantity cosine 5 pi over 6 plus i times sine 5 pi over 6 end quantity question mark

Answer 1

Given the expression:

`6(cos(5\pi)/(6)+i\sin (5\pi)/(6))`

You can identify that it has this form of a Complex Number:

`a+bi`

Where "a" and "b" are Real Numbers.

• By definition, you can rewrite:

`\cos (5\pi)/(6)`

in this form:

`\cos (\pi-(\pi)/(6))`

Simplifying using this Trigonometric Identity:

`\cos (\pi-x)=-\cos x`

Then:

`-\cos (\pi)/(6)`

By definition, its value is:

`-\cos (\pi)/(6)=-(√(3))/(2)`

• Use the same reasoning for:

`\sin (5\pi)/(6)`

Using this Trigonometric Identity:

`\sin (\pi-x)=\sin x`

You get:

`\sin (5\pi)/(6)=\sin (\pi-(\pi)/(6))=\sin (\pi)/(6)`

By definition:

`\sin (\pi)/(6)=(1)/(2)`

• Therefore, you can rewrite the expression as follows:

`=6(-\frac{\sqrt[]{3}}{2}+(1)/(2)i)`

Apply the Distributive Property and simplify:

`\begin{gathered} =-\frac{6\sqrt[]{3}}{2}+(6)/(2)i \\ \\ =-3\sqrt[]{3}+3i \end{gathered}`

• In order to identify which point is represented by the Complex Number, you need to identify the value that corresponds to the Real axis. This is:

`-3\sqrt[]{3}\approx-5.2`

And the value that corresponds to the Imaginary Axis:

`3`

Notice that the point with those coordinates in the Complex Plane is:

`Q(-5.2,3)`

Hence, the answer is: First option.