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For exercise 1:

Given:
\[
\cos x = \frac{1}{2}, \quad \frac{3\pi}{2} < x < 2\pi
\]

First, determine the value of \(x\) within the given range. The value \(x\) that satisfies \(\cos x = \frac{1}{2}\) in the interval \(\frac{3\pi}{2} < x < 2\pi\) is:
\[
x = \frac{5\pi}{3}
\]

Next, calculate \(x/2\):
\[
x/2 = \frac{5\pi}{6}
\]

Now, use the half-angle formulas to calculate \(\sin(x/2)\), \(\cos(x/2)\), and \(\tan(x/2)\):

1. \(\sin \frac{x}{2}\):
\[
\sin \frac{5\pi}{6} = \sin 150^\circ = \frac{1}{2}
\]

2. \(\cos \frac{x}{2}\):
\[
\cos \frac{5\pi}{6} = \cos 150^\circ = -\frac{\sqrt{3}}{2}
\]

3. \(\tan \frac{x}{2}\):
\[
\tan \frac{5\pi}{6} = \tan 150^\circ = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}
\]

Thus, the results are:
\[
\sin \frac{x}{2} = \frac{1}{2}, \quad \cos \frac{x}{2} = -\frac{\sqrt{3}}{2}, \quad \tan \frac{x}{2} = -\frac{\sqrt{3}}{3}
\]

If you have any questions or need further details, please let me know!

Here are five additional questions you can ask:
1. How do I find the value of \(\sin(x + y)\) using the sum formula?
2. Can you solve another trigonometric identity problem?
3. How do I derive the double angle formulas?
4. What are the key steps in solving trigonometric equations?
5. How can I verify trigonometric identities?

**Tip:** When working with trigonometric identities and equations, always consider the quadrant in which the angle lies to determine the correct signs of the trigonometric functions.

Learn how to solve a trigonometric equation involving cos x = 1/2 and calculate half-angle formulas step-by-step. Understand the concepts of trigonometry and apply half-angle formulas to find sin(x/2), cos(x/2), and tan(x/2). Suitable for high school students and those preparing for trigonometry exams.

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