Problem 48 - Precalculus Essentials

Question

Find the exact value of each trigonometric function. Do not use a calculator. $$-\cot \left(\frac{\pi}{4}+17 \pi\right)$$

Step-by-Step Solution

Answer

The exact value of \( - \cot \left(\frac{\pi}{4}+17 \pi\right) \) is -1.

Step 1: Simplify the Argument of Cotangent

Simplify the argument inside the cotangent function. The cotangent function has periodicity of \( \pi \). So, \( \cot \left(\frac{\pi}{4}+17 \pi\right) = \cot \left(\frac{\pi}{4}\right) \).

Step 2: Find the Value of Cotangent

Now, substitute \( \frac{\pi}{4} \) into the cotangent function. Use the fact that the cotangent is the reciprocal of tangent. The value of \( \tan(\frac{\pi}{4}) \) is 1, so the cotangent of \( \frac{\pi}{4} \) is \( \frac{1}{\tan(\frac{\pi}{4})} = 1 \).

Step 3: Apply the Negative Sign

Now apply the negative sign outside the cotangent function. So, \( - \cot \left(\frac{\pi}{4}+17 \pi\right) = -1 \).