Problem 52 - Precalculus Essentials

Question

Find the exact value of each trigonometric function. Do not use a calculator. $$\cos \left(-\frac{\pi}{4}-2000 \pi\right)$$

Step-by-Step Solution

Answer

\(\cos \left(-\frac{\pi}{4}-2000 \pi\right) = \cos (\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\)

Step 1: Understanding the Cosine Function

The cosine function, \(\cos (x)\), is a periodic function with a period of \(2\pi\). This means that for any integer \(n\), \(\cos (x) = \cos (x + 2n\pi)\). You can therefore simplify the term \(-\frac{\pi}{4}-2000 \pi\).

Step 2: Simplify the Term

Based on the periodic property of the cosine function, \(\cos \left(-\frac{\pi}{4}-2000 \pi\right) = \cos (-\frac{\pi}{4} - 2000 \pi + 2000*2\pi) = \cos (-\frac{\pi}{4})\).

Step 3: Use the Cosine of Negative Angle Formula

Now, apply the cosine of a negative angle formula: \(\cos (-x) = \cos (x)\). Hence, \(\cos (-\frac{\pi}{4}) = \cos (\frac{\pi}{4})\).