https://r2.erweima.ai/mathbot/image/1721212761491uyw3zhtn.jpg

To solve exercise 1 from the uploaded image, we need to determine the values of \(\sin \frac{x}{2}\), \(\cos \frac{x}{2}\), and \(\tan \frac{x}{2}\) given that \(\cos x = \frac{1}{2}\) and \(\frac{3\pi}{2} < x < 2\pi\).

First, let's identify the value of \(x\):
Since \(\cos x = \frac{1}{2}\) and \(x\) is in the interval \(\frac{3\pi}{2} < x < 2\pi\), \(x\) is in the fourth quadrant. The reference angle for \(\cos x = \frac{1}{2}\) is \(\frac{\pi}{3}\). Thus, \(x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}\).

Now, let's calculate the required trigonometric functions:

### a) \(\sin \frac{x}{2}\)

Using the half-angle formula for sine:
\[
\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}
\]
Since \(x = \frac{5\pi}{3}\), \(\frac{x}{2} = \frac{5\pi}{6}\) lies in the second quadrant where sine is positive. Therefore:
\[
\sin \frac{x}{2} = \sqrt{\frac{1 - \frac{1}{2}}{2}} = \sqrt{\frac{1}{4}} = \frac{1}{2}
\]

### b) \(\cos \frac{x}{2}\)

Using the half-angle formula for cosine:
\[
\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}
\]
Since \(\frac{x}{2} = \frac{5\pi}{6}\) lies in the second quadrant where cosine is negative:
\[
\cos \frac{x}{2} = -\sqrt{\frac{1 + \frac{1}{2}}{2}} = -\sqrt{\frac{3}{4}} = -\frac{\sqrt{3}}{2}
\]

### c) \(\tan \frac{x}{2}\)

Using the half-angle formula for tangent:
\[
\tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}
\]
Since \(\frac{x}{2} = \frac{5\pi}{6}\) lies in the second quadrant where tangent is negative:
\[
\tan \frac{x}{2} = -\sqrt{\frac{1 - \frac{1}{2}}{1 + \frac{1}{2}}} = -\sqrt{\frac{\frac{1}{2}}{\frac{3}{2}}} = -\sqrt{\frac{1}{3}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}
\]

So, the solutions are:
1. a) \(\sin \frac{x}{2} = \frac{1}{2}\)
2. b) \(\cos \frac{x}{2} = -\frac{\sqrt{3}}{2}\)
3. c) \(\tan \frac{x}{2} = -\frac{\sqrt{3}}{3}\)

Would you like details on these calculations or have any questions? Here are 5 more questions you can ask:

1. How do you solve exercise 2 from the uploaded image?
2. What is the solution for exercise 3 from the uploaded image?
3. Can you explain the steps for solving exercise 4?
4. How do you approach exercise 5 from the image?
5. What are the solutions for exercise 6 in the image?

**Tip:** When solving trigonometric problems, always determine the quadrant to ensure the correct sign for the trigonometric values.

Learn how to solve Exercise 1 from a trigonometry problem involving finding the values of sin, cos, and tan of half angles given specific conditions. Detailed step-by-step solution provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation