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Chapter 1: Problem 6

Evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(f(x)=\sin x\) (a) \(f(\pi)\) (b) \(f(5 \pi / 4)\) (c) \(f(2 \pi / 3)\)

Short Answer

Expert verified
The results for the function evaluations are: (a) 0, (b) \(-\sqrt{2}/2\), and (c) \(\sqrt{3}/2\).

Step by step solution

01

Evaluating \(f(\pi)\)

We use the fact that \( \sin \pi = 0 \). Thus, \( f(\pi) = \sin \pi = 0 \).
02

Evaluating \(f(5\pi / 4)\)

To find \( f(5\pi / 4) \), we use the fact that the sine of any angle \( \theta \) in the fourth quadrant, i.e., \( \frac{3\pi}{2} \leq \theta \leq 2\pi \), is negative. First, we can find an equivalent angle in the first quadrant by subtracting \( 2\pi \) or adding multiples of \( 2\pi \), until we land in the first quadrant. In this case, \( \theta_1 = 5\pi/4 - \pi = \pi/4 \) lies in the first quadrant. And we know that \( \sin(\pi/4) = \sqrt{2}/2 \). Hence, \( f(5\pi / 4) = -\sin(\pi / 4) = -\sqrt{2}/2 \).
03

Evaluating \(f(2\pi / 3)\)

The given \( x \) value \(2\pi / 3\) falls in the second quadrant where sin function returns positive values. We know that \( \sin(2\pi / 3) \) will have the same absolute value as \( \sin(\pi / 3) \) because both angles add up to \( \pi \). We know that \( \sin(\pi / 3) = \sqrt{3}/2 \), so \( f(2\pi / 3) = \sqrt{3}/2 \).

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