Find the values of all the the T-Rations if :-
`" sin theta =(5)/(13)"`

Answer

To find the values of all the trigonometric ratios given that \( \sin \theta = \frac{5}{13} \), we can follow these steps:

### Step 1: Understand the definition of sine
We know that:
\[ \sin \theta = \frac{\text{perpendicular}}{\text{hypotenuse}} \]
In this case, we have:
\[ \sin \theta = \frac{5}{13} \]
This means the perpendicular side of the right triangle is 5 units and the hypotenuse is 13 units.

### Step 2: Draw the right triangle
Draw a right triangle where:
- The angle \( \theta \) is one of the angles.
- The side opposite to \( \theta \) (perpendicular) is 5 units.
- The hypotenuse is 13 units.

### Step 3: Calculate the base (adjacent side)
Using the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{perpendicular}^2 + \text{base}^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + \text{base}^2 \]
This simplifies to:
\[ 169 = 25 + \text{base}^2 \]
\[ \text{base}^2 = 169 - 25 = 144 \]
Taking the square root:
\[ \text{base} = \sqrt{144} = 12 \]

### Step 4: Calculate the remaining trigonometric ratios
Now that we have all three sides of the triangle:
- Perpendicular (opposite) = 5
- Base (adjacent) = 12
- Hypotenuse = 13

We can find the other trigonometric ratios:

1. **Cosine**:
\[ \cos \theta = \frac{\text{base}}{\text{hypotenuse}} = \frac{12}{13} \]

2. **Tangent**:
\[ \tan \theta = \frac{\text{perpendicular}}{\text{base}} = \frac{5}{12} \]

3. **Cosecant** (reciprocal of sine):
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{13}{5} \]

4. **Secant** (reciprocal of cosine):
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{13}{12} \]

5. **Cotangent** (reciprocal of tangent):
\[ \cot \theta = \frac{1}{\tan \theta} = \frac{12}{5} \]

### Final Values of Trigonometric Ratios:
- \( \sin \theta = \frac{5}{13} \)
- \( \cos \theta = \frac{12}{13} \)
- \( \tan \theta = \frac{5}{12} \)
- \( \csc \theta = \frac{13}{5} \)
- \( \sec \theta = \frac{13}{12} \)
- \( \cot \theta = \frac{12}{5} \)