The cosecant of a certain angle is `(13)/(12)`. Evaluate other t-ratios of this angle.

Answer

To solve the problem step by step, we will evaluate the trigonometric ratios based on the given cosecant value.

### Step 1: Identify the given value
We are given that:
\[ \csc \theta = \frac{13}{12} \]

### Step 2: Understand the relationship of cosecant
Recall that:
\[ \csc \theta = \frac{1}{\sin \theta} \]
Thus, we can find \(\sin \theta\):
\[ \sin \theta = \frac{1}{\csc \theta} = \frac{12}{13} \]

### Step 3: Use the Pythagorean theorem
In a right triangle, the relationship between the sides is given by:
\[ \text{hypotenuse}^2 = \text{perpendicular}^2 + \text{base}^2 \]
Here, the hypotenuse is 13 (from cosecant), and the perpendicular (opposite side) is 12 (from sine). We need to find the base (adjacent side).

Let the base be \(b\). Then:
\[ 13^2 = 12^2 + b^2 \]
Calculating the squares:
\[ 169 = 144 + b^2 \]
Subtracting 144 from both sides:
\[ b^2 = 169 - 144 = 25 \]
Taking the square root:
\[ b = 5 \]

### Step 4: Calculate other trigonometric ratios
Now we have:
- Hypotenuse = 13
- Perpendicular = 12
- Base = 5

1. **Sine**:
\[ \sin \theta = \frac{\text{perpendicular}}{\text{hypotenuse}} = \frac{12}{13} \]

2. **Cosine**:
\[ \cos \theta = \frac{\text{base}}{\text{hypotenuse}} = \frac{5}{13} \]

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

4. **Cosecant** (already given):
\[ \csc \theta = \frac{13}{12} \]

5. **Secant**:
\[ \sec \theta = \frac{\text{hypotenuse}}{\text{base}} = \frac{13}{5} \]

6. **Cotangent**:
\[ \cot \theta = \frac{\text{base}}{\text{perpendicular}} = \frac{5}{12} \]

### Summary of all trigonometric ratios:
- \(\sin \theta = \frac{12}{13}\)
- \(\cos \theta = \frac{5}{13}\)
- \(\tan \theta = \frac{12}{5}\)
- \(\csc \theta = \frac{13}{12}\)
- \(\sec \theta = \frac{13}{5}\)
- \(\cot \theta = \frac{5}{12}\)