What is the technical solution of this equation ?

To find the technical solution of the equation:

\[
\cos(x) = \frac{1}{2} \sqrt{2 + \sqrt{3}}
\]

we follow these steps:

### Step 1: Consider the equation
\[
\cos(x) = \frac{1}{2} \sqrt{2 + \sqrt{3}}
\]

We need to find the angles \(x\) such that the cosine of \(x\) equals this value.

### Step 2: Analyze the expression
Given that \( \cos(x) \) values range between \(-1\) and \(1\), the expression \( \frac{1}{2} \sqrt{2 + \sqrt{3}} \) must lie within this range. 

Let's evaluate \( \frac{1}{2} \sqrt{2 + \sqrt{3}} \) to check if it is indeed within \([-1, 1]\).

\[
2 + \sqrt{3} \approx 2 + 1.732 = 3.732
\]

\[
\sqrt{3.732} \approx 1.931
\]

\[
\frac{1.931}{2} \approx 0.9655
\]

Since \(0.9655\) is within \([-1, 1]\), the equation is valid.

### Step 3: Express the solution
Since \( \cos(x) \) is a periodic function with a period of \(2\pi\), there are generally two solutions for \(x\) within any \(2\pi\) interval. 

If \( \cos(x) = A \), then:

\[
x_1 = \cos^{-1}(A)
\]

\[
x_2 = 2\pi - \cos^{-1}(A)
\]

Applying this to our specific case:

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

or

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

### Step 4: General solution
Since the cosine function is periodic, the general solutions are given by:

\[
x = 2n\pi \pm \cos^{-1}\left(\frac{\sqrt{2 + \sqrt{3}}}{2}\right), \quad n \in \mathbb{Z}
\]

where \( n \) is any integer.

### Conclusion
The technical solution to the equation \( \cos(x) = \frac{1}{2} \sqrt{2 + \sqrt{3}} \) is:

\[
x = 2n\pi \pm \cos^{-1}\left(\frac{\sqrt{2 + \sqrt{3}}}{2}\right), \quad n \in \mathbb{Z}
\]

This accounts for all possible values of \( x \) that satisfy the equation.

Let me know if you need further details or if you have any questions!

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Here are 5 related questions:
1. How can you find the exact value of \( \cos^{-1} \) for special angles?
2. What are the properties of inverse trigonometric functions?
3. How do you solve equations involving inverse trigonometric functions?
4. Can we express \( \frac{\sqrt{2 + \sqrt{3}}}{2} \) in a simpler form using trigonometric identities?
5. How does the periodicity of trigonometric functions affect the number of solutions?

**Tip:** When dealing with inverse trigonometric functions, always consider the principal value range of the function and the periodicity of the trigonometric function for the general solution.

Explore the technical solution for the equation \( \cos(x) = \frac{1}{2} \sqrt{2 + \sqrt{3}} \) using inverse trigonometric functions. Learn how to find all possible values of \( x \) that satisfy the equation, considering the periodic nature of trigonometric functions.

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