4sin^2x-3=0 on 0,2pi answer in terms of pi/radians

To solve the equation \( 4\sin^2(x) - 3 = 0 \) over the interval \( [0, 2\pi] \), follow these steps:

1. **Isolate the trigonometric function:**
   \[
   4\sin^2(x) = 3
   \]
   \[
   \sin^2(x) = \frac{3}{4}
   \]

2. **Take the square root of both sides:**
   \[
   \sin(x) = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}
   \]

3. **Determine the general solution:**
   The sine function equals \( \frac{\sqrt{3}}{2} \) at specific angles:
   \[
   x = \frac{\pi}{3}, \quad x = \pi - \frac{\pi}{3} = \frac{2\pi}{3}
   \]
   The sine function equals \( -\frac{\sqrt{3}}{2} \) at specific angles:
   \[
   x = \pi + \frac{\pi}{3} = \frac{4\pi}{3}, \quad x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}
   \]

4. **List all solutions within the interval \( [0, 2\pi] \):**
   The solutions to the equation in this interval are:
   \[
   x = \frac{\pi}{3}, \quad x = \frac{2\pi}{3}, \quad x = \frac{4\pi}{3}, \quad x = \frac{5\pi}{3}
   \]

### Final Answer:
The solutions to the equation \( 4\sin^2(x) - 3 = 0 \) in the interval \( [0, 2\pi] \) are:
\[
x = \frac{\pi}{3}, \quad x = \frac{2\pi}{3}, \quad x = \frac{4\pi}{3}, \quad x = \frac{5\pi}{3}
\]

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### Related Questions:
1. How do you solve trigonometric equations involving cosine?
2. What is the general solution for \( \sin(x) = \frac{1}{2} \)?
3. How do you solve a quadratic equation in terms of sine or cosine?
4. How do you determine the exact values of sine and cosine for standard angles?
5. What are the properties of the sine and cosine functions over \( [0, 2\pi] \)?

### Tip:
Remember that \( \sin(x) = \frac{\sqrt{3}}{2} \) and \( \sin(x) = -\frac{\sqrt{3}}{2} \) correspond to specific angles that are common on the unit circle, which makes solving such equations easier.

Learn how to solve the trigonometric equation 4sin^2(x) - 3 = 0 over the interval [0, 2π] with detailed step-by-step solutions. This problem involves applying trigonometric identities and understanding the Pythagorean identity. Suitable for students in Grades 10-12.

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