Question

use reference angles to find the exact value of the following expression. tan(9π/4) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. tan(9π/4) = (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize the denominator.) b. the answer is undefined.

Explanation:

Step1: Find a coterminal angle.

Subtract \(2\pi\) from \(\frac{9\pi}{4}\) since \(2\pi\) is one full - rotation. \(\frac{9\pi}{4}-2\pi=\frac{9\pi - 8\pi}{4}=\frac{\pi}{4}\).

Step2: Determine the reference angle.

The angle \(\frac{\pi}{4}\) is in the first quadrant, and for an angle \(\theta=\frac{\pi}{4}\) in the first quadrant, the reference angle \(\theta_{r}=\theta=\frac{\pi}{4}\).

Step3: Evaluate the tangent function.

We know that \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and for \(\theta = \frac{\pi}{4}\), \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\). So \(\tan\frac{\pi}{4}=\frac{\sin\frac{\pi}{4}}{\cos\frac{\pi}{4}}=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1\).

Answer:

A. \(\tan\frac{9\pi}{4}=1\)