Question

use reference angles to find the exact value of the following expression. do not use a calculator.
\\( \sin \left( - 300 ^ { \circ } \
ight) \\)
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\( \bigcirc \\) a. \\( \sin \left( - 300 ^ { \circ } \
ight) = \\)
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize the denominator.)
\\( \bigcirc \\) b. the answer is undefined.

Explanation:

Step1: Use the odd - even property of sine

The sine function is an odd function, which means that $\sin(-\theta)=-\sin(\theta)$. So, $\sin(-300^{\circ})=-\sin(300^{\circ})$.

Step2: Find the reference angle of $300^{\circ}$

To find the reference angle of an angle $\theta$ in the fourth quadrant (where $270^{\circ}<\theta < 360^{\circ}$), the reference angle $\alpha$ is given by $\alpha = 360^{\circ}-\theta$. For $\theta = 300^{\circ}$, the reference angle $\alpha=360^{\circ}- 300^{\circ}=60^{\circ}$.

Step3: Determine the sign of $\sin(300^{\circ})$

In the fourth quadrant, the sine function is negative (since sine corresponds to the y - coordinate on the unit circle, and in the fourth quadrant, $y<0$). So, $\sin(300^{\circ})=-\sin(60^{\circ})$.

Step4: Recall the value of $\sin(60^{\circ})$

We know that $\sin(60^{\circ})=\frac{\sqrt{3}}{2}$.

Step5: Calculate $\sin(-300^{\circ})$

From Step1, $\sin(-300^{\circ})=-\sin(300^{\circ})$. From Step3, $\sin(300^{\circ})=-\sin(60^{\circ})$, so $\sin(-300^{\circ})=-(-\sin(60^{\circ}))=\sin(60^{\circ})=\frac{\sqrt{3}}{2}$.

Answer:

A. $\sin(-300^{\circ}) = \frac{\sqrt{3}}{2}$