We are solving for \( x \) in the equation:

\[
\tan(x) = -\frac{\sqrt{3}}{2}
\]

To solve this, let's first break it down:

### Step 1: Understanding the Tangent Value
The tangent function represents the ratio of the opposite side to the adjacent side in a right triangle. However, \(\tan(x)\) values like \(\frac{\sqrt{3}}{2}\) do not correspond to any standard angle in radians or degrees.

Tangent values typically follow recognizable forms like:
\[
\tan(30^\circ) = \frac{1}{\sqrt{3}}, \quad \tan(45^\circ) = 1, \quad \tan(60^\circ) = \sqrt{3}
\]
But \(-\frac{\sqrt{3}}{2}\) is not a standard value from these, meaning this specific tangent does not correspond to a common angle.

### Step 2: Calculator Solution
Since \(\frac{\sqrt{3}}{2}\) is not a standard trigonometric value, we can use the inverse tangent function (\(\tan^{-1}\)) on a calculator to solve for \( x \). However, because the tangent function is periodic with a period of \( \pi \), the general solution will also account for this.

Using a calculator for \(\tan^{-1}\left(-\frac{\sqrt{3}}{2}\right)\), we get an approximate solution for \( x \):

\[
x \approx -0.9828 \, \text{radians} \quad (\text{principal value})
\]

### Step 3: General Solution
Since the tangent function has a period of \( \pi \), the general solution is:

\[
x = -0.9828 + n\pi \quad \text{where} \, n \in \mathbb{Z}
\]

Thus, \( x \) could take on infinitely many values spaced by \( \pi \).

### Summary
The general solution for the equation \( \tan(x) = -\frac{\sqrt{3}}{2} \) is:

\[
x = -0.9828 + n\pi \quad \text{for any integer} \, n.
\]

Would you like more details on how to find the general solution for periodic trigonometric functions, or have any questions? 

Here are 5 related questions to expand your understanding:
1. What is the periodicity of the tangent function, and how does it affect solutions?
2. How do you solve trigonometric equations with non-standard values?
3. What are the inverse trigonometric functions, and how are they used?
4. How does the range of

Learn how to solve the equation tan(x) = -√3/2 with step-by-step instructions. This problem covers key trigonometric concepts, including tangent functions, inverse trigonometry, and periodicity. Suitable for high school students in Grades 10-12.

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