Given the expression:
[tex]6(cos\frac{5\pi}{6}+i\sin \frac{5\pi}{6})[/tex]You can identify that it has this form of a Complex Number:
[tex]a+bi[/tex]Where "a" and "b" are Real Numbers.
• By definition, you can rewrite:
[tex]\cos \frac{5\pi}{6}[/tex]in this form:
[tex]\cos (\pi-\frac{\pi}{6})[/tex]Simplifying using this Trigonometric Identity:
[tex]\cos (\pi-x)=-\cos x[/tex]Then:
[tex]-\cos \frac{\pi}{6}[/tex]By definition, its value is:
[tex]-\cos \frac{\pi}{6}=-\frac{\sqrt{3}}{2}[/tex]• Use the same reasoning for:
[tex]\sin \frac{5\pi}{6}[/tex]Using this Trigonometric Identity:
[tex]\sin (\pi-x)=\sin x[/tex]You get:
[tex]\sin \frac{5\pi}{6}=\sin (\pi-\frac{\pi}{6})=\sin \frac{\pi}{6}[/tex]By definition:
[tex]\sin \frac{\pi}{6}=\frac{1}{2}[/tex]• Therefore, you can rewrite the expression as follows:
[tex]=6(-\frac{\sqrt[]{3}}{2}+\frac{1}{2}i)[/tex]Apply the Distributive Property and simplify:
[tex]\begin{gathered} =-\frac{6\sqrt[]{3}}{2}+\frac{6}{2}i \\ \\ =-3\sqrt[]{3}+3i \end{gathered}[/tex]• 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:
[tex]-3\sqrt[]{3}\approx-5.2[/tex]And the value that corresponds to the Imaginary Axis:
[tex]3[/tex]Notice that the point with those coordinates in the Complex Plane is:
[tex]Q(-5.2,3)[/tex]Hence, the answer is: First option.