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derive the triangle area formula using trigonometric ratios

To derive the area of triangle \( \triangle ABC \) using trigonometric ratios, let's start with the information given in the diagram.

### Given:
- \( \theta \) is the angle at \( C \).
- \( a = BC \) (opposite side of \( \theta \)).
- \( b = CA \) (adjacent side to \( \theta \)).
- \( h = \) height from \( B \) perpendicular to \( CA \).

### Goal:
Derive the formula for the area of \( \triangle ABC \) in terms of \( a \), \( b \), and \( \theta \).

### Step 1: Express the area using base and height.
The area \( A \) of the triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is \( b \), and the height is \( h \). So,
\[
A = \frac{1}{2} \times b \times h
\]

### Step 2: Express the height \( h \) using trigonometry.
In the right triangle \( \triangle BCA \), the height \( h \) can be expressed using the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{a}
\]
Thus,
\[
h = a \sin(\theta)
\]

### Step 3: Substitute \( h \) into the area formula.
Now, substitute \( h = a \sin(\theta) \) into the area formula:
\[
A = \frac{1}{2} \times b \times a \sin(\theta)
\]

### Final Formula:
\[
\boxed{A = \frac{1}{2} ab \sin(\theta)}
\]
This is the formula for the area of a triangle using the lengths of two sides \( a \) and \( b \) and the sine of the included angle \( \theta \).

Would you like further details, or do you have any questions?

Here are some related questions you might find interesting:
1. How can you derive the area formula for a triangle using other trigonometric functions like cosine or tangent?
2. How does the area formula change if \( \theta \) is a right angle?
3. What happens to the area formula if one of the sides of the triangle is the hypotenuse of a right triangle?
4. Can the formula \( A = \frac{1}{2} ab \sin(\theta) \) be used for non-acute triangles?
5. How can you derive the area of a triangle using Heron's formula, and how does it compare to this method?

**Tip:** When dealing with trigonometric ratios, always ensure the angle and sides correspond correctly in the right triangle. This avoids errors in calculations or derivations.

Learn how to derive the formula for the area of a triangle using trigonometric ratios. This problem covers key concepts in trigonometry and geometry, including the use of sine function to calculate the area of a triangle. Suitable for high school students studying trigonometry.

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