Problem 63 - Precalculus Essentials

Question

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\tan 210^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \(\tan 210^{\circ}\) is \(\frac{\sqrt{3}}{3}\).

Step 1: Identify Quadrant

First, identify which quadrant the given angle falls into. An angle of 210 degrees is beyond 180 degrees but less than 270 degrees. Therefore, it falls into the third quadrant of the unit circle.

Step 2: Find Reference Angle

The reference angle is the acute angle that the given angle makes with the x-axis. To find the reference angle for an angle in the third quadrant, subtract the angle from 180 degrees. Hence, \(180^{\circ} - 210^{\circ}= -30^{\circ}\). The reference angle is thus 30 degrees.

Step 3: Apply Tangent Function

In the third quadrant, the tangent function is positive. Thus, we simply find the tangent of the reference angle, which is \(\tan 30^{\circ} = \frac{\sqrt{3}}{3}\).