26 Multiple Answer 1 point
Solve the following equation over the interval . Select all of the solutions.

Ask by Norris Nunez. in the United States

Mar 28,2025

Question

26 Multiple Answer 1 point Solve the following equation over the interval $$ [0,2 \pi) $$. Select all of the solutions. $$ (\cot x-1)(2 \cos x-\sqrt{3}) $$

UpStudy AI · Accepted Answer

We start with

$$
(\cot x-1)(2 \cos x-\sqrt{3})=0.
$$

Since a product is zero if and only if at least one factor is zero, we have two cases.

**Case 1:** $$\cot x - 1 = 0$$

This simplifies to

$$
\cot x = 1.
$$

Recall that $$\cot x = \frac{\cos x}{\sin x}$$. The equation $$\cot x = 1$$ implies

$$
\frac{\cos x}{\sin x} = 1 \quad \implies \quad \cos x = \sin x.
$$

The equality $$\cos x = \sin x$$ holds for

$$
x = \frac{\pi}{4} \quad \text{and} \quad x = \frac{5\pi}{4}
$$

within the interval $$[0, 2\pi)$$. (Note that at $$x = 0, \pi,2\pi$$, $$\sin x$$ is zero and cotangent is undefined, so these are not included.)

**Case 2:** $$2 \cos x - \sqrt{3} = 0$$

Solving for $$\cos x$$ we get

$$
2 \cos x = \sqrt{3} \quad \implies \quad \cos x = \frac{\sqrt{3}}{2}.
$$

On the interval $$[0,2\pi)$$, the solutions for $$\cos x = \frac{\sqrt{3}}{2}$$ are

$$
x = \frac{\pi}{6} \quad \text{and} \quad x = \frac{11\pi}{6}.
$$

**Final Answer:**

The solutions to the equation on $$[0,2\pi)$$ are

$$
x = \frac{\pi}{4},\quad \frac{5\pi}{4},\quad \frac{\pi}{6},\quad \text{and} \quad \frac{11\pi}{6}.
$$
The solutions are $$ x = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{\pi}{6}, $$ and $$ \frac{11\pi}{6} $$.

26 Multiple Answer 1 point
Solve the following equation over the interval . Select all of the solutions.

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26 Multiple Answer 1 point Solve the following equation over the interval . Select all of the solutions.