Problem 50 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The exact value of the given trigonometric function is -\(\sqrt{2}/2\).

Step 1: Simplify the Inside of Sin Function

Observe that there is a -2000\(\pi\) included in the angle given inside the sin function. It is known from periodic properties of sin function that its value repeats after every 2\(\pi\). It means that you can ignore any multiple of 2\(\pi\) in the expression \(-\pi/4 - 2000\pi\), this simplifies to \(-\pi/4\).

Step 2: Find the Sin value

The value of sin function for negative angle -\(\pi/4\) is same as for \(\pi/4\). But remember to keep the negative sign. \(\sin(-\pi/4) = -\sin(\pi/4)\)

Step 3: Using the value of \(\sin(\pi/4)\)

The value of \(\sin(\pi/4)\) is known to be \(\sqrt{2}/2\). So, \(\sin(-\pi/4) = -\sqrt{2}/2\).