Problem 71
Question
In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\tan \frac{9 \pi}{4}$$
Step-by-Step Solution
Verified Answer
The exact value of \( \tan \frac{9 \pi}{4} \) is 1.
1Step 1: Understanding the periodicity of the tan function
Since the tangent function has a periodicity of \(\pi\), every integer multiple of \(\pi\) would effectively bring us back to the same point. Hence, we can subtract any multiple of \(\pi\) from our given angle until we get an equivalent angle that is within our standard unit circle. Thus, \(\tan \frac{9 \pi}{4} = \tan \frac{8 \pi}{4} + \frac{\pi}{4} = \tan \frac{\pi}{4}\).
2Step 2: Calculate the tangent of the simplified angle
The angle \( \frac{\pi}{4} \) represents a 45-degree angle, and it is known that \( \tan(45°) = 1 \). So, \( \tan \frac{\pi}{4} = 1 \).
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