Which triogonometric ratio related the adjacent side and opposite side of the right-angled triangle?

Options

A. `sintheta`
B. `costheta`
C. `tantheta`
D. `sectheta`

Answer

To determine which trigonometric ratio relates the adjacent side and opposite side of a right-angled triangle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Right-Angled Triangle**:
- Let's denote the triangle as \( \triangle ABC \) where \( \angle B \) is the right angle.

2. **Label the Sides**:
- Let \( AB \) be the opposite side to angle \( \theta \).
- Let \( BC \) be the adjacent side to angle \( \theta \).

3. **Define the Angle**:
- We will consider \( \theta \) as one of the angles in the triangle, specifically \( \angle A \).

4. **Identify the Trigonometric Ratios**:
- The trigonometric ratios for a right-angled triangle are:
- Sine (\( \sin \)): Opposite / Hypotenuse
- Cosine (\( \cos \)): Adjacent / Hypotenuse
- Tangent (\( \tan \)): Opposite / Adjacent

5. **Focus on the Ratio of Interest**:
- We need to find the ratio that relates the adjacent side \( BC \) and the opposite side \( AB \).

6. **Determine the Correct Ratio**:
- The ratio that relates the opposite side \( AB \) to the adjacent side \( BC \) is the tangent function:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]

7. **Conclusion**:
- Therefore, the trigonometric ratio that relates the adjacent side and the opposite side of a right-angled triangle is called the tangent ratio, denoted as \( \tan(\theta) \).

### Final Answer:
The trigonometric ratio that relates the adjacent side and opposite side of a right-angled triangle is **Tangent (tan)**.

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