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 Express sin A cos C into the sum of trigonometric functions using the following triangle.

Given the right triangle, we are tasked with expressing \( \sin A \cdot \cos C \) in terms of the sum of trigonometric functions.

### Step 1: Identify the angles
From the triangle provided:
- \( \angle A = 60^\circ \)
- \( \angle B = 90^\circ \) (right angle)
- \( \angle C = 30^\circ \)

### Step 2: Express \( \sin A \) and \( \cos C \)
1. **Sine of angle \( A \)**:
   \[
   \sin A = \sin 60^\circ = \frac{\sqrt{3}}{2}
   \]

2. **Cosine of angle \( C \)**:
   \[
   \cos C = \cos 30^\circ = \frac{\sqrt{3}}{2}
   \]

### Step 3: Compute \( \sin A \cdot \cos C \)
\[
\sin A \cdot \cos C = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3}{4}
\]

### Step 4: Express as a sum of trigonometric functions
We can express the product of sine and cosine using the following identity:
\[
\sin A \cos C = \frac{1}{2} \left[ \sin(A + C) + \sin(A - C) \right]
\]

Substituting \( A = 60^\circ \) and \( C = 30^\circ \):
\[
\sin 60^\circ \cos 30^\circ = \frac{1}{2} \left[ \sin(60^\circ + 30^\circ) + \sin(60^\circ - 30^\circ) \right]
\]

### Step 5: Simplify the expression
Now calculate the sums inside the sine functions:
\[
\sin(90^\circ) = 1 \quad \text{and} \quad \sin(30^\circ) = \frac{1}{2}
\]

So:
\[
\sin 60^\circ \cos 30^\circ = \frac{1}{2} \left[ 1 + \frac{1}{2} \right] = \frac{1}{2} \cdot \frac{3}{2} = \frac{3}{4}
\]

### Final Expression:
\[
\sin A \cos C = \frac{1}{2} \left[ \sin(90^\circ) + \sin(30^\circ) \right] = \frac{3}{4}
\]

Do you want further details or have any questions?

Here are some related questions:
1. How can the cosine of angles \( A \) and \( B \) be calculated?
2. What is the sum \( \sin C \cdot \cos A \) for this triangle?
3. How would the expression change if \( \angle C \) was different?
4. What is the relationship between \( \sin(A + C) \) and \( \cos(A - C) \)?
5. Can \( \sin A \cdot \cos C \) be expressed using a different identity?
6. What are the values of \( \tan A \) and \( \tan C \) in this triangle?
7. How would the area of the triangle be calculated if a side length is given?
8. What is the geometric significance of the angles in this triangle?

**Tip:** Always check the sum of angles in a triangle to ensure consistency when dealing with trigonometric identities.

Learn how to express sin A cos C using trigonometric identities in a right triangle. Understand the steps involved and solve similar trigonometric problems. Suitable for high school students studying trigonometry.

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