Question
Solve the following inequality below for all values on the interval . Use interval notation to represent the solution set in the answer s)
below.

Ask by Ryan Tyler. in the United States

Mar 27,2025

Question

Question
Solve the following inequality below for all values on the interval . Use interval notation to represent the solution set in the answer s)
below.

Ask by Ryan Tyler. in the United States

Mar 27,2025

UpStudy AI · Accepted Answer

We start with the inequality

$$
4 \cos ^{2} \theta - 1 \leq 0.
$$

**Step 1. Isolate the cosine squared term**

Add 1 to both sides:

$$
4 \cos^2 \theta \leq 1.
$$

Now, divide both sides by 4:

$$
\cos^2 \theta \leq \frac{1}{4}.
$$

**Step 2. Take the square root**

This inequality is equivalent to

$$
\left| \cos \theta \right| \leq \frac{1}{2}.
$$

which means

$$
-\frac{1}{2} \leq \cos \theta \leq \frac{1}{2}.
$$

**Step 3. Find the critical angles**

We know that

$$
\cos \theta = \frac{1}{2} \quad \text{at} \quad \theta = \frac{\pi}{3} \text{ and } \theta = \frac{5\pi}{3},
$$
$$
\cos \theta = -\frac{1}{2} \quad \text{at} \quad \theta = \frac{2\pi}{3} \text{ and } \theta = \frac{4\pi}{3}.
$$

**Step 4. Determine the intervals where the inequality holds**

The inequality $$\left| \cos \theta \right| \leq \frac{1}{2}$$ means that the cosine value is between $$-\frac{1}{2}$$ and $$\frac{1}{2}$$. Over the interval $$0 \leq \theta \leq 2\pi$$ this occurs in the following two arcs:

- From $$\theta = \frac{\pi}{3}$$ to $$\theta = \frac{2\pi}{3}$$, where the cosine decreases from $$\frac{1}{2}$$ to $$-\frac{1}{2}$$.
- From $$\theta = \frac{4\pi}{3}$$ to $$\theta = \frac{5\pi}{3}$$, where the cosine increases from $$-\frac{1}{2}$$ to $$\frac{1}{2}$$.

**Step 5. Write the final answer in interval notation**

Thus, the solution set is

$$
\left[\frac{\pi}{3}, \frac{2\pi}{3}\right] \cup \left[\frac{4\pi}{3}, \frac{5\pi}{3}\right].
$$
The solution to the inequality $$4 \cos^{2} \theta - 1 \leq 0$$ for $$0 \leq \theta \leq 2\pi$$ is: $$ \left[\frac{\pi}{3}, \frac{2\pi}{3}\right] \cup \left[\frac{4\pi}{3}, \frac{5\pi}{3}\right] $$

Question
Solve the following inequality below for all values on the interval . Use interval notation to represent the solution set in the answer s)
below.

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Question Solve the following inequality below for all values on the interval . Use interval notation to represent the solution set in the answer s) below.

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Question
Solve the following inequality below for all values on the interval . Use interval notation to represent the solution set in the answer s)
below.