Solve the equation for all solutions in the interval [0, 27) .
Cos (x/2) = = -((sqrt 3)/2)
x = ?

To solve the equation \( \cos\left(\frac{x}{2}\right) = -\frac{\sqrt{3}}{2} \), we will follow a systematic approach:

### Step 1: Determine the angle(s) for which the cosine is \( -\frac{\sqrt{3}}{2} \).
The cosine of an angle is \( -\frac{\sqrt{3}}{2} \) at the angles:

\[
\theta = \frac{5\pi}{6} \quad \text{and} \quad \theta = \frac{7\pi}{6}
\]

These are the standard angles in radians on the unit circle where the cosine takes the value \( -\frac{\sqrt{3}}{2} \). However, since \( \frac{x}{2} = \theta \), we have to find all possible angles within the interval \( [0, 27) \).

### Step 2: Express solutions for \( x/2 \).
Since the cosine function is periodic with a period of \( 2\pi \), the general solution for \( \frac{x}{2} \) will be:

\[
\frac{x}{2} = \frac{5\pi}{6} + 2k\pi \quad \text{or} \quad \frac{x}{2} = \frac{7\pi}{6} + 2k\pi
\]

where \( k \) is any integer.

### Step 3: Solve for \( x \).
To solve for \( x \), multiply both sides of the equations by 2:

\[
x = \frac{10\pi}{6} + 4k\pi \quad \text{or} \quad x = \frac{14\pi}{6} + 4k\pi
\]
\[
x = \frac{5\pi}{3} + 4k\pi \quad \text{or} \quad x = \frac{7\pi}{3} + 4k\pi
\]

### Step 4: Find solutions within the interval \( [0, 27) \).
Now, we will substitute integer values for \( k \) to find all solutions for \( x \) in the interval \( [0, 27) \).

#### For \( x = \frac{5\pi}{3} + 4k\pi \):

- When \( k = 0 \): \( x = \frac{5\pi}{3} \approx 5.24 \)
- When \( k = 1 \): \( x = \frac{5\pi}{3} + 4\pi = \frac{5\pi}{3} + 12.57 \approx 5.24 + 12.57 = 17.81 \)
- When \( k = 2 \): \( x = \frac{5\pi}{3} + 8\pi = \frac{5\pi}{3} + 25.13 \approx 5.24 + 25.13 = 30.37 \) (This is outside the interval.)

So, the valid solutions from this branch are \( x \approx 5.24 \) and \( x \approx 17.81 \).

#### For \( x = \frac{7\pi}{3} + 4k\pi \):

- When \( k = 0 \): \( x = \frac{7\pi}{3} \approx 7.33 \)
- When \( k = 1 \): \( x = \frac{7\pi}{3} + 4\pi = \frac{7\pi}{3} + 12.57 \approx 7.33 + 12.57 = 19.90 \)
- When \( k = 2 \): \( x = \frac{7\pi}{3} + 8\pi = \frac{7\pi}{3} + 25.13 \approx 7.33 + 25.13 = 32.46 \) (This is outside the interval.)

So, the valid solutions from this branch are \( x \approx 7.33 \) and \( x \approx 19.90 \).

### Final Solutions:
The solutions for \( x \) in the interval \( [0, 27) \) are:

\[
x \approx 5.24, 7.33, 17.81, 19.90
\]

Would you like further details on any part of this process, or do you have any questions?

Here are 5 related questions to explore:

1. How do you find the general solution of a trigonometric equation?
2. What is the period of the cosine function?
3. Why are there two standard angles for a given cosine value?
4. How do you convert between degrees and radians?
5. What is the significance of the unit circle in trigonometry?

**Tip:** Always keep track of the function's period when solving trigonometric equations, as it helps identify all possible solutions within the desired interval.

Learn how to solve the trigonometric equation cos(x/2) = -sqrt(3)/2 step-by-step. This problem involves trigonometric functions and their periodicity. Suitable for advanced high school students and anyone preparing for calculus.

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