Problem 65 - Precalculus Essentials

Question

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\tan 420^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \( \tan 420^{\circ} \) is \( \sqrt{3} \).

Step 1: Find the Coterminal Angle

Since 420 degrees is more than one full rotation (360 degrees), subtract 360 degrees from 420 degrees to find a coterminal angle within the first rotation. So, \(420^{\circ} - 360^{\circ} = 60^{\circ}\). Our coterminal angle is then 60 degrees.

Step 2: Determine the Quadrant and the Reference Angle

60 degrees lies in the first quadrant, where the tangent of an angle is positive. Also, since angles in the first quadrant are their own reference angles, the reference angle is also 60 degrees.

Step 3: Obtain the Tangent Value

Next, the value of the tangent of 60 degrees should be obtained. This is a commonly known value. We have \(\tan 60^{\circ} = \sqrt{3}\).