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Mark the trigonometric ratios in this photo 

In the provided image, the left diagram represents a triangle inscribed in a semicircle, which allows us to identify and label trigonometric ratios. Let's break this down:

1. **Triangle OAB**:
   - \( \angle AOB \) is a right angle (since it lies on the diameter of the circle, following the property of a semicircle).
   - Let \( \angle BOA = \theta \).

### **Trigonometric Ratios**:
For triangle OAB, we can label the sides in terms of trigonometric ratios with respect to \( \angle \theta \):

- **Opposite side** to \( \theta \): Side \( AB \) (denoted by \( a \)).
- **Adjacent side** to \( \theta \): Side \( OB \) (denoted by \( b \)).
- **Hypotenuse**: Side \( OA \) (denoted by \( c \)).

Now, we can define the standard trigonometric ratios:

- **Sine**: \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{OA} \)
- **Cosine**: \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{OB}{OA} \)
- **Tangent**: \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{OB} \)

Similarly, we can write the reciprocal functions:

- **Cosecant**: \( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{OA}{AB} \)
- **Secant**: \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{OA}{OB} \)
- **Cotangent**: \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{OB}{AB} \)

### Short Note on Trigonometric Ratios:
Trigonometric ratios are fundamental in understanding the relationships between the angles and sides of a right triangle. For any acute angle \( \theta \), the ratios \( \sin(\theta), \cos(\theta), \) and \( \tan(\theta) \) give the relative proportions of two sides of the triangle. These ratios are extensively used in various applications such as calculating heights, distances, and solving problems in physics and engineering.

Let me know if you need further clarification or additional details!

### Relative questions:
1. How do trigonometric ratios change as the angle \( \theta \) increases?
2. Can we define trigonometric ratios for angles greater than \( 90^\circ \)?
3. What is the geometric significance of the reciprocal trigonometric functions?
4. How does the Pythagorean identity relate to the diagram of the triangle?
5. How can we apply trigonometric ratios to real-world problems, such as in navigation or architecture?

**Tip**: When solving problems involving trigonometric ratios, always ensure that you correctly identify the opposite, adjacent, and hypotenuse sides with respect to the angle in question.

Observe the given figures and label the trigonometric ratios. Also, write a short note.

Learn how to label trigonometric ratios (sine, cosine, tangent, and their reciprocals) using a triangle inscribed in a semicircle. This problem explains the relationship between the angle and sides of a right triangle with clear formulas and theorems. Suitable for students in Grades 9-11.

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