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Let's solve both questions one by one.

### Question 1 (36): 
**Given**: 
- \( \cos(\theta) = \frac{1}{2} \)
- \( \theta \) is in the first quadrant.
  
We need to find \( \sin\left(\frac{\theta}{2}\right) \) using a half-angle identity.

**Solution**:
The half-angle identity for sine is:

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

Since \( \theta \) is in the first quadrant, \( \frac{\theta}{2} \) will also be in the first quadrant, so the sine will be positive.

1. Substitute \( \cos(\theta) = \frac{1}{2} \):

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

Thus, the correct answer is **B**.

---

### Question 2 (37):
We need to identify the correct half-angle identity for cosine.

The half-angle identity for cosine is:

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

Thus, the correct option is **A**.

---

Would you like a more detailed explanation on any of the steps? Here are 5 related questions that you might want to explore next:

1. What is the half-angle identity for tangent?
2. How do we determine the sign of the half-angle identity in other quadrants?
3. How are half-angle identities derived from trigonometric identities?
4. Can you solve for \( \cos\left(\frac{\theta}{2}\right) \) using the given information?
5. What are other trigonometric identities used in calculus?

**Tip**: Always check the quadrant in which your angle lies when applying half-angle identities, as it determines whether the result will be positive or negative.

Using a half-angle identity, find sin(θ/2) if cos(θ) = 1/2 and θ is in the first quadrant. Which of the following is the half-angle identity for cosine?

Learn how to find sin(θ/2) using half-angle identities when cos(θ) = 1/2, and determine the half-angle identity for cosine. This guide includes step-by-step solutions and key trigonometric formulas, ideal for high school students studying trigonometry.

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