Problem 46 - Precalculus Essentials

Question

Find the exact value of each trigonometric function. Do not use a calculator. $$\tan \left(-\frac{9 \pi}{4}\right)$$

Step-by-Step Solution

Answer

The exact value of \( \tan \left(-\frac{9 \pi}{4}\right)\) is -1.

Step 1: Simplify the Angle

Given the angle \(-\frac{9\pi}{4}\), this angle can be simplified using the periodicity property of the tangent function. The tangent function has a period of \(\pi\), which means that \(\tan(\theta) = \tan(\theta + n\pi)\) for any integer n. This property allows us to add \(\pi\) to the -9π/4 multiple times until we get an angle that is easier to deal with. It would be best to add \(2\pi\) which is an equivalent of a full circle, since doing so won't affect the tangent's value. Adding \(2\pi\) is the same as adding \(8\pi/4\), so \(- \frac{9\pi}{4} + \frac{8\pi}{4} = - \frac{\pi}{4}\).

Step 2: Evaluate the Tangent Function

Now, it's left to find the tangent of the simplified angle. So, based on the unit circle, the tangent of -π/4 equals -1, because \(\tan(-\frac{\pi}{4}) = -1\).