Problem 49 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The exact value of the function \( \sin( -\pi/4 - 1000\pi ) \) is \( -\frac{1}{\sqrt{2}} \).

Step 1: Analyze the function and exploit periodicity

Given the function sin (-π/4 - 1000π). We observe that 1000π is a multiple of 2π. Sine being a function with a period of 2π, addition or subtraction of any multiple of 2π does not affect its value as it corresponds to complete cycles on the unit circle. Hence, sin(-π/4 - 1000π) = sin(-π/4).

Step 2: Calculate the value

From trigonometric values for standard angles, we know that \( \sin(\pi/4) = \frac{1}{\sqrt{2}} \). Since sine function is symmetrical with respect to origin, sin(-π/4) is the negative of sin(π/4). Therefore, \( \sin(-\pi/4) = -\frac{1}{\sqrt{2}} \). Hence, \( \sin(-\pi/4 - 1000\pi) = -\frac{1}{\sqrt{2}} \).