Question

use reference angles to find the exact value of the following expression. do not use a calculator.
sin(-240°)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. sin(-240°)=□
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.

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, for \(\sin(-240^{\circ})\), we can rewrite it as \(\sin(-240^{\circ})=-\sin(240^{\circ})\).

Step2: Find the reference angle of \(240^{\circ}\)

To find the reference angle of an angle in the third quadrant (\(180^{\circ}<\theta < 270^{\circ}\)), we use the formula \(\text{Reference angle}=\theta - 180^{\circ}\). For \(\theta = 240^{\circ}\), the reference angle \(\alpha=240^{\circ}-180^{\circ} = 60^{\circ}\).

Step3: Determine the sign of \(\sin(240^{\circ})\)

In the third quadrant, the sine function is negative (since \(y\) - coordinates are negative in the third quadrant). So, \(\sin(240^{\circ})=-\sin(60^{\circ})\).

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

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

Step5: Calculate \(\sin(-240^{\circ})\)

From Step 1, \(\sin(-240^{\circ})=-\sin(240^{\circ})\). From Step 3, \(\sin(240^{\circ})=-\sin(60^{\circ})\), so \(\sin(-240^{\circ})=-(-\sin(60^{\circ}))=\sin(60^{\circ})\). Substituting the value of \(\sin(60^{\circ})\), we get \(\sin(-240^{\circ})=\frac{\sqrt{3}}{2}\).

Answer:

\(\frac{\sqrt{3}}{2}\)