Problem 66 - Precalculus Essentials

Question

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\tan 405^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \(\tan 405^{\circ}\) is \(1\).

Step 1: Converting the angle into the range of \(0-360^{\circ}\)

To convert \(405^{\circ}\) into its equivalent value in the range of \(0-360^{\circ}\), \(405-360=45\) degrees is calculated to get \(45^{\circ}\), which is within the defined range.

Step 2: Determine the reference angle

A reference angle is positive acute angle that can be associated with any angle. Since \(45^{\circ}\) is an acute angle itself, it serves as its own reference angle. Thus, the reference angle is \(45^{\circ}\).

Step 3: Find the value of the trigonometric function

Knowing the reference angle, it can now be substituted into the trigonometric function to find its exact value. For this exercise it is known that \(\tan 45^{\circ} = 1\). Therefore, \(\tan 405^{\circ} = \tan 45^{\circ} = 1\).