Problem 45 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The exact value of \( \cot(-5pi/4) \) is -1.

Step 1: Identify the Quadrant of the Angle

The given angle is -5pi/4. As the negative sign indicates, it's measured clockwise from the positive x-axis. Simply neglecting the negative sign, when we check 5pi/4, it is more than pi but less than 2pi, this indicates that the angle is in 3rd quadrant. Since it's a negative angle, we move clockwise direction to reach the 3rd quadrant.

Step 2: Find the corresponding angle in the First Quadrant

The corresponding angle in the positive direction for -5pi/4 can be calculated as |(-5pi/4) mod (2pi)|, which is pi/4. Note that for any angle \( \theta \), \(\cot(-\theta) = -\cot(\theta)\) due to the property of the cotangent function.

Step 3: Compute the cotangent of the angle

The cotangent of an angle \( \theta \) in quadrant 3 is negative, but we just found that our corresponding angle was \( \theta \) = pi/4. So, the cotangent will be determined as \( -\cot(pi/4) \). We know that \( \cot(pi/4) = 1 \), since the x and y coordinates of point on the unit circle for pi/4 are equal.

Step 4: Write the final answer

Combining the information from the above steps, we get the final solution as \( -\cot(pi/4) = -1 \)