If `cot theta=(4)/(3)`, find the values of other t-ratios of `theta`.

Answer

To find the values of the other trigonometric ratios of \( \theta \) given that \( \cot \theta = \frac{4}{3} \), we can follow these steps:

### Step 1: Understand the relationship of cotangent
The cotangent function is defined as the ratio of the adjacent side (base) to the opposite side (perpendicular) in a right triangle:
\[ \cot \theta = \frac{\text{base}}{\text{perpendicular}} = \frac{b}{p} \]
Given \( \cot \theta = \frac{4}{3} \), we can assign:
- Base \( b = 4 \)
- Perpendicular \( p = 3 \)

### Step 2: Use the Pythagorean theorem to find the hypotenuse
To find the hypotenuse \( h \), we use the Pythagorean theorem:
\[ h = \sqrt{p^2 + b^2} \]
Substituting the values of \( p \) and \( b \):
\[ h = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Calculate the other trigonometric ratios
Now that we have all three sides of the triangle, we can calculate the other trigonometric ratios:

1. **Sine**:
\[ \sin \theta = \frac{p}{h} = \frac{3}{5} \]

2. **Cosine**:
\[ \cos \theta = \frac{b}{h} = \frac{4}{5} \]

3. **Tangent**:
\[ \tan \theta = \frac{p}{b} = \frac{3}{4} \]

4. **Cosecant** (reciprocal of sine):
\[ \csc \theta = \frac{h}{p} = \frac{5}{3} \]

5. **Secant** (reciprocal of cosine):
\[ \sec \theta = \frac{h}{b} = \frac{5}{4} \]

6. **Cotangent** (already given):
\[ \cot \theta = \frac{b}{p} = \frac{4}{3} \]

### Summary of Trigonometric Ratios
Thus, the values of the trigonometric ratios are:
- \( \sin \theta = \frac{3}{5} \)
- \( \cos \theta = \frac{4}{5} \)
- \( \tan \theta = \frac{3}{4} \)
- \( \csc \theta = \frac{5}{3} \)
- \( \sec \theta = \frac{5}{4} \)
- \( \cot \theta = \frac{4}{3} \)