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Given the following information, use a half angle identity to find the exact value of the following expression.


To solve the problem using the half-angle identity, we'll follow these steps:

### Step 1: Identify the given information
Given:
- \(\cos \theta = -\frac{3}{4}\)
- \(90^\circ < \theta < 180^\circ\) (which means \(\theta\) is in the second quadrant)

### Step 2: Use the half-angle identity
The half-angle identity for cosine is given by:

\[
\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos \theta}{2}}
\]

The sign is determined by the quadrant in which \(\frac{\theta}{2}\) lies. Since \(90^\circ < \theta < 180^\circ\), \(\frac{\theta}{2}\) lies in the first quadrant, where cosine is positive.

### Step 3: Substitute the given value of \(\cos \theta\)
Substitute \(\cos \theta = -\frac{3}{4}\) into the half-angle identity:

\[
\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \left(-\frac{3}{4}\right)}{2}} = \sqrt{\frac{1 - \frac{3}{4}}{2}}
\]

Simplify the expression:

\[
\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{\frac{1}{4}}{2}} = \sqrt{\frac{1}{8}} = \frac{\sqrt{1}}{\sqrt{8}} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}
\]

### Step 4: Match with the options
The value of \(\cos\left(\frac{\theta}{2}\right)\) is \(\frac{\sqrt{2}}{4}\), which corresponds to option D.

**Answer:** The correct option is **D**.

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Would you like more details on any part of this solution or have any further questions?

Here are some related questions you might find useful:
1. How does the sign of the half-angle identity change depending on the quadrant?
2. What is the derivation of the half-angle identity for cosine?
3. How do you apply half-angle identities in integration problems?
4. What are some common errors when using half-angle identities?
5. Can you derive the sine half-angle identity using the cosine half-angle identity?

**Tip:** Always ensure the correct sign for trigonometric identities based on the quadrant of the angle in question.

Learn how to find the exact value of a trigonometric expression using the half-angle identity for cosine. This problem involves trigonometric concepts and the application of identities in solving for specific angles. Suitable for high school students studying trigonometry.

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