Problem 43 - Precalculus Essentials

Question

Find the exact value of each trigonometric function. Do not use a calculator. $$\tan \frac{5 \pi}{4}$$

Step-by-Step Solution

Answer

The exact value of \(\tan(\frac{5 \pi}{4})\) is 1.

Step 1: Understand the Unit circle

The unit circle is a circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system. The angle formed by the positive x-axis and the line connecting the origin to a point P on the circle represents the measure of that angle. In this case, with the angle being \(\frac{5 \pi}{4}\), we are more than halfway around the circle, specifically \(\frac{5}{4}\) turns, which is in the 3rd quadrant of the unit circle.

Step 2: Determine the equivalent angle in the first turn

While \(\frac{5 \pi}{4}\) spins us more than full way around the circle, we are actually ending in the same position as if we only spun \(\frac{1 \pi}{4}\) the way around the circle. The reason is that the trigonometric functions are periodic in nature. This makes our equivalent angle be \(\frac{1 \pi}{4}\).

Step 3: Find the value of the tangent

The tangent is defined as the ratio of the y-coordinate to the x-coordinate of the point P on the unit circle. In the third quadrant, the sign of the tangent function is positive, as in the first quadrant. The value of \(\tan(\frac{\pi}{4})\) is 1, so the value of \(\tan(\frac{5 \pi}{4})\) is also 1.