Problem 44 - Precalculus Essentials

Question

Find the exact value of each trigonometric function. Do not use a calculator. $$\cot \frac{5 \pi}{4}$$

Step-by-Step Solution

Answer

The exact value of \( \cot \frac{5 \pi}{4} \) is 1.

Step 1: Convert Radians to Degrees

Convert the measured in radians angle \(\frac{5\pi}{4}\) to degrees: \(\frac{5\pi}{4} \times \frac{180}{\pi}\) = 225 degrees. Think of the unit circle, this angle is in the third quadrant.

Step 2: Find Sin and Cos Values

Recall that in the unit circle, a 45 degrees angle in the third quadrant has the sine and cosine values as -1/√2 or -√2/2.

Step 3: Calculate Cotangent

The trigonometric function cotangent is the reciprocal of the tangent function, and the tangent function is the ratio of sine to cosine. So, cotangent = cos/sin = -√2/2/ (-√2/2) = 1