If `sec theta = (5)/(3) and 0 lt theta lt (pi)/(2)`. Find all the other T-ratios.

Explanation

To solve the problem, we will use the information given about the secant function and the properties of a right triangle.

### Step-by-Step Solution:

1. **Understanding Secant**:
We know that secant (sec) is defined as the reciprocal of cosine (cos). Therefore, if \( \sec \theta = \frac{5}{3} \), then:
\[ \cos \theta = \frac{1}{\sec \theta} = \frac{3}{5} \]

2. **Drawing a Right Triangle**:
We can visualize this with a right triangle where:
- The adjacent side (base) to angle \( \theta \) is 3.
- The hypotenuse is 5.

3. **Finding the Opposite Side**:
To find the length of the opposite side (perpendicular), we can use the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{adjacent}^2 + \text{opposite}^2 \]
Plugging in the values:
\[ 5^2 = 3^2 + \text{opposite}^2 \]
\[ 25 = 9 + \text{opposite}^2 \]
\[ \text{opposite}^2 = 25 - 9 = 16 \]
\[ \text{opposite} = 4 \]

4. **Finding Other Trigonometric Ratios**:
Now that we have all sides of the triangle, we can find the other trigonometric ratios:
- **Sine**:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5} \]
- **Tangent**:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3} \]
- **Cosecant** (csc):
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{5}{4} \]
- **Cotangent** (cot):
\[ \cot \theta = \frac{1}{\tan \theta} = \frac{3}{4} \]

5. **Summary of All Trigonometric Ratios**:
- \( \sin \theta = \frac{4}{5} \)
- \( \cos \theta = \frac{3}{5} \)
- \( \tan \theta = \frac{4}{3} \)
- \( \csc \theta = \frac{5}{4} \)
- \( \sec \theta = \frac{5}{3} \)
- \( \cot \theta = \frac{3}{4} \)