https://file.aiquickdraw.com/mathbot/file/17254872223040waiszcf.jpg

The problem appears to involve a trigonometric equation. I will analyze the image and extract the problem to solve it.

The equation in the image is:

\[
\sin^2(x) - \frac{3}{4}\cos(x) - \frac{1}{2} = 0
\]

To solve this, let's proceed step by step:

1. **Rewrite the equation in terms of one trigonometric function**: Since \(\sin^2(x) = 1 - \cos^2(x)\), substitute this into the equation.

\[
(1 - \cos^2(x)) - \frac{3}{4}\cos(x) - \frac{1}{2} = 0
\]

2. **Expand and simplify**:

\[
1 - \cos^2(x) - \frac{3}{4}\cos(x) - \frac{1}{2} = 0
\]

Combine like terms:

\[
-\cos^2(x) - \frac{3}{4}\cos(x) + \frac{1}{2} = 0
\]

Multiply the entire equation by -1 to make it easier to solve:

\[
\cos^2(x) + \frac{3}{4}\cos(x) - \frac{1}{2} = 0
\]

3. **Solve for \(\cos(x)\) using the quadratic formula**: 

The equation is in the standard quadratic form \(a\cos^2(x) + b\cos(x) + c = 0\), where \(a = 1\), \(b = \frac{3}{4}\), and \(c = -\frac{1}{2}\).

The quadratic formula is:

\[
\cos(x) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute the values of \(a\), \(b\), and \(c\):

\[
\cos(x) = \frac{-\frac{3}{4} \pm \sqrt{\left(\frac{3}{4}\right)^2 - 4 \cdot 1 \cdot \left(-\frac{1}{2}\right)}}{2}
\]

Let's calculate the solutions for \(\cos(x)\).The solutions for \(\cos(x)\) are:

\[
\cos(x) = 0.425
\]
\[
\cos(x) = -1.175
\]

However, the value \(\cos(x) = -1.175\) is not possible since the range of the cosine function is between \(-1\) and \(1\). Therefore, the only valid solution is:

\[
\cos(x) = 0.425
\]

4. **Find the general solution for \(x\)**:

Since \(\cos(x) = 0.425\), the general solution for \(x\) is:

\[
x = \cos^{-1}(0.425) + 2k\pi \quad \text{or} \quad x = 2\pi - \cos^{-1}(0.425) + 2k\pi
\]

where \(k\) is any integer. Let's compute these values for one cycle (\(0 \leq x < 2\pi\)).The solutions for \(x\) within one cycle (\(0 \leq x < 2\pi\)) are:

\[
x_1 \approx 1.132 \, \text{radians}
\]
\[
x_2 \approx 5.151 \, \text{radians}
\]

These are the values of \(x\) that satisfy the given equation.

### Would you like more details or have any questions?

### Related Questions:
1. How can you solve a quadratic equation involving sine and cosine simultaneously?
2. What are the trigonometric identities useful for converting between sine and cosine?
3. How do we handle equations when the trigonometric function results fall outside their typical ranges?
4. What are the general forms of solutions for trigonometric equations?
5. How do you graph the solutions of trigonometric equations to verify them visually?

### Tip:
When solving trigonometric equations, always verify that the solutions fall within the range of the trigonometric function and the desired domain (e.g., \(0 \leq x < 2\pi\)).

Solve the trigonometric equation: sin²(x) - (3/4)cos(x) - 1/2 = 0.

Learn how to solve the trigonometric equation sin²(x) - (3/4)cos(x) - 1/2 = 0 with a step-by-step solution. This problem involves key trigonometric identities and quadratic equations, suitable for students in Grades 10-12.

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