Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \tan \left( \frac{15\pi}{4} \right) = -1 \)
Solution
Calculate or simplify the expression \( \tan(\frac{15\pi}{4}) \).
Calculate the value by following steps:
- step0: Calculate:
\(\tan\left(\frac{15\pi }{4}\right)\)
- step1: Rearrange the numbers:
\(\tan\left(\frac{3\pi }{4}+3\pi \right)\)
- step2: Rearrange the terms:
\(\tan\left(\frac{3\pi }{4}\right)\)
- step3: Rewrite the expression:
\(-\tan\left(\frac{\pi }{4}\right)\)
- step4: Calculate:
\(-1\)
The value of \( \tan \left( \frac{15\pi}{4} \right) \) is -1.
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Simplify this solution Bonus Knowledge
To find \( \tan \frac{15 \pi}{4} \), we first need to simplify the angle. The tangent function has a period of \( \pi \), which means that \( \tan(x) = \tan(x + n\pi) \) for any integer \( n \). Let's reduce \( \frac{15 \pi}{4} \): \[ \frac{15 \pi}{4} - 4\pi = \frac{15 \pi}{4} - \frac{16 \pi}{4} = -\frac{\pi}{4} \] Now we can find \( \tan(-\frac{\pi}{4}) \). The tangent of \( -\frac{\pi}{4} \) is: \[ \tan(-\frac{\pi}{4}) = -\tan(\frac{\pi}{4}) = -1 \] So, \( \tan \frac{15 \pi}{4} = -1 \).
