Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Oct 24, 2023

Find the values of $x$ between $0$ and $4\pi$ where $\cos x = \frac{1}{2}$ . List the answers in ascending order.

STEP 1

Assumptions1. We are working with the function $\cos x$ . . We are looking for values of $x$ between $x=0$ and $x=4 \pi$ .
3. We are looking for values of $x$ for which $\cos x=\frac{1}{}$ .
4. We know that the cosine function is periodic with period $\pi$ .
5. We know that $\cos x = \frac{1}{}$ for $x = \frac{\pi}{3}$ and $x = \frac{5\pi}{3}$ in the interval $0 \leq x <\pi$ .

STEP 2

Since the cosine function is periodic with period $2\pi$ , the solutions to the equation $\cos x=\frac{1}{2}$ in the interval $0 \leq x <2\pi$ will repeat every $2\pi$ .

STEP 3

We know that $\cos x = \frac{1}{2}$ for $x = \frac{\pi}{3}$ and $x = \frac{5\pi}{3}$ in the interval $0 \leq x <2\pi$ .

STEP 4

We can find the solutions in the interval $2\pi \leq x <4\pi$ by adding $2\pi$ to each of the solutions in the interval $0 \leq x <2\pi$ .
$x = \frac{\pi}{3} +2\pi, \frac{\pi}{3} +2\pi$

STEP 5

Calculate the solutions in the interval $2\pi \leq x <4\pi$ .
$x = \frac{\pi}{3} +2\pi = \frac{7\pi}{3}, \frac{5\pi}{3} +2\pi = \frac{11\pi}{3}$

STEP 6

Combine the solutions from the intervals $0 \leq x <2\pi$ and $2\pi \leq x <4\pi$ to find all solutions in the interval $0 \leq x <4\pi$ .
$x = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{\pi}{3}, \frac{11\pi}{3}$ So the values of $x$ between $x=0$ and $x=4 \pi$ for which $\cos x=\frac{1}{2}$ are $x = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{\pi}{3}, \frac{11\pi}{3}$ .

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