Solve the equation. Give a general formula for all the solutions. List six solutions.

Identify the general formula for all the solutions to based on the smaller angle,
is an integer
(Simplify your answer. Use angle measures greater than or equal to 0 and less than . Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression. Type
an expression using k as the variable.)

Ask by Joseph Wade. in the United States

Jan 20,2025

Question

Solve the equation. Give a general formula for all the solutions. List six solutions.

Identify the general formula for all the solutions to based on the smaller angle,
is an integer
(Simplify your answer. Use angle measures greater than or equal to 0 and less than . Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression. Type
an expression using k as the variable.)

Ask by Joseph Wade. in the United States

Jan 20,2025

UpStudy AI · Accepted Answer

To solve the equation:

$$
\sin \theta = \frac{\sqrt{3}}{2}
$$

we seek all angles $$\theta$$ that satisfy this equation. The sine function equals $$\frac{\sqrt{3}}{2}$$ at specific standard angles within the interval $$[0, 2\pi)$$.

### Step 1: Identify the Reference Angles

The value $$\frac{\sqrt{3}}{2}$$ corresponds to the sine of $$\frac{\pi}{3}$$ radians (or 60 degrees). Since the sine function is positive in both the first and second quadrants, the two primary solutions within one period $$[0, 2\pi)$$ are:

$$
\theta = \frac{\pi}{3} \quad \text{and} \quad \theta = \frac{2\pi}{3}
$$

### Step 2: Generalize the Solutions

The sine function has a period of $$2\pi$$, meaning the solutions repeat every $$2\pi$$ radians. Therefore, the general solutions can be expressed as:

$$
\theta = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad \theta = \frac{2\pi}{3} + 2k\pi \quad \text{for any integer } k
$$

### Step 3: List Six Specific Solutions

To provide six specific solutions within the interval $$[0, 4\pi)$$, we can consider two full periods ($$0$$ to $$4\pi$$):

1. For $$k = 0$$:
   - $$\theta = \frac{\pi}{3}$$
   - $$\theta = \frac{2\pi}{3}$$

2. For $$k = 1$$:
   - $$\theta = \frac{\pi}{3} + 2\pi = \frac{7\pi}{3}$$
   - $$\theta = \frac{2\pi}{3} + 2\pi = \frac{8\pi}{3}$$

3. For $$k = 2$$:
   - $$\theta = \frac{\pi}{3} + 4\pi = \frac{13\pi}{3}$$
   - $$\theta = \frac{2\pi}{3} + 4\pi = \frac{14\pi}{3}$$

### Summary

**General Solution:**

$$
\theta = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad \theta = \frac{2\pi}{3} + 2k\pi \quad \text{for any integer } k
$$

**Six Specific Solutions:**

$$
\theta = \frac{\pi}{3}, \quad \frac{2\pi}{3}, \quad \frac{7\pi}{3}, \quad \frac{8\pi}{3}, \quad \frac{13\pi}{3}, \quad \frac{14\pi}{3}
$$
The general solution for $$\sin \theta = \frac{\sqrt{3}}{2}$$ is: $$ \theta = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad \theta = \frac{2\pi}{3} + 2k\pi \quad \text{for any integer } k $$ Six specific solutions within $$[0, 4\pi)$$ are: $$ \theta = \frac{\pi}{3}, \quad \frac{2\pi}{3}, \quad \frac{7\pi}{3}, \quad \frac{8\pi}{3}, \quad \frac{13\pi}{3}, \quad \frac{14\pi}{3} $$

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