https://file.aiquickdraw.com/mathbot/file/1729474601768203ony91.jpg

We are tasked with solving the trigonometric equation from the image, which is:

\[
\sqrt{3} \sec\left(\frac{1}{2}x\right) + 2 = 0
\]

We will solve this step-by-step for \( x \) within the range \( -11 < x < 11 \).

### Step 1: Isolate the secant term

First, subtract 2 from both sides of the equation:

\[
\sqrt{3} \sec\left(\frac{1}{2}x\right) = -2
\]

### Step 2: Solve for the secant function

Now, divide both sides of the equation by \( \sqrt{3} \):

\[
\sec\left(\frac{1}{2}x\right) = -\frac{2}{\sqrt{3}}
\]

Simplifying \( -\frac{2}{\sqrt{3}} \), we rationalize the denominator:

\[
\sec\left(\frac{1}{2}x\right) = -\frac{2\sqrt{3}}{3}
\]

### Step 3: Inverse secant to cosine

Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \), so:

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

### Step 4: Solve for \( \frac{1}{2}x \)

The cosine of \( \frac{1}{2}x \) is \( -\frac{\sqrt{3}}{2} \). The angle where cosine equals \( -\frac{\sqrt{3}}{2} \) is:

\[
\frac{1}{2}x = \frac{5\pi}{6}, \frac{7\pi}{6} + 2k\pi \quad (k \in \mathbb{Z})
\]

### Step 5: Solve for \( x \)

Multiply both sides of these equations by 2:

\[
x = \frac{5\pi}{3}, \frac{7\pi}{3} + 4k\pi
\]

### Step 6: Convert to numerical values and check the range \( -11 < x < 11 \)

Using the approximate value of \( \pi \approx 3.1416 \):

- \( x = \frac{5\pi}{3} \approx 5.24 \)
- \( x = \frac{7\pi}{3} \approx 7.33 \)

Now, adjust by adding multiples of \( 4\pi \) (approximately 12.57) to find any other solutions that fit within the given range \( -11 < x < 11 \).

### Step 7: Final solutions

The solutions within \( -11 < x < 11 \) are:

\[
x \approx 5.24, \quad x \approx 7.33
\]

Would you like more details on any of these steps, or help with a related concept? Here are some related questions to consider:

1. How can you graph secant and cosine functions to visualize these solutions?
2. What is the general solution form for equations involving secant and cosine?
3. How do phase shifts in trigonometric functions affect the solutions?
4. How do you solve trigonometric equations that involve more complex expressions?
5. How can we check for extraneous solutions in trigonometric equations?

**Tip:** Always rationalize denominators when simplifying trigonometric identities for clarity and consistency in solutions.

Solve the trigonometric equation for all values −11 < x < 11. √3 sec(1/2)x + 2 = 0.

Learn how to solve the trigonometric equation √3 sec(1/2)x + 2 = 0 for x within the range −11 < x < 11. This detailed guide covers the secant function, cosine inverse theorem, and key trigonometric identities. Suitable for undergraduate students.

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