Problem 73 - Precalculus Essentials

Question

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\sin \left(-240^{\circ}\right)$$

Step-by-Step Solution

Answer

The exact value of \(\sin(-240^{\circ})\) is \(\sqrt{3}/2\).

Step 1: Convert the angle to a positive measure

An angle of \(-240^{\circ}\) actually moves in the clockwise direction. To convert it to a positive measure, add \(360^{\circ}\) to it. Therefore, the angle measured in the counterclockwise direction will be \( 360^{\circ}-240^{\circ}=120^{\circ}\)

Step 2: Find the reference angle

A reference angle is always computed relative to the x-axis in the clockwise direction. For an angle of \(120^{\circ}\), the reference angle would be \( 180^{\circ}-120^{\circ}=60^{\circ}\)

Step 3: Use the reference angle to find exact sine value

The angle \(120^{\circ}\) lies in the 2nd quadrant where sine is positive. The Sine of \(60^{\circ}\) is \(\sqrt{3}/2\). Therefore, \(\sin(-240^{\circ}) = \sin(120^{\circ}) = \sin(60^{\circ}) = \sqrt{3}/2\).