Problem 51 - Precalculus Essentials

Question

Find the exact value of each trigonometric function. Do not use a calculator. $$\cos \left(-\frac{\pi}{4}-1000 \pi\right)$$

Step-by-Step Solution

Answer

The exact value of the trigonometric function \(\cos \left(-\frac{\pi}{4}-1000 \pi\right)\) is \(\frac{\sqrt{2}}{2}\).

Step 1: Simplify the angle

Firstly, simplify the angle in the function \(\cos( -\frac{\pi}{4} - 1000\pi )\). Notice that \(1000\pi\) is a multiple of \(2\pi\), hence it has no effect on the value of the cosine function. Therefore, the function simplifies to \(\cos( -\frac{\pi}{4})\).

Step 2: Evaluate the cosine function

Cosine function is symmetric in the y-axis. Therefore, cos(\(\theta\)) = cos(-\(\theta\)). Hence, we can simplify \(\cos( -\frac{\pi}{4})\) to \(\cos( \frac{\pi}{4})\). The exact value of \(\cos( \frac{\pi}{4})\) is \(\frac{\sqrt{2}}{2}\).