If `sin theta = (5)/(13),` then the values of `tan theta and sec theta ` respectively, are

Options

A. `(5)/(12), (13)/(12)`
B. `(11)/(13), (5)/(12)`
C. `(14)/(19), (5)/(12)`
D. `(5)/(13), (5)/(12)`

Answer

To solve the problem, we need to find the values of \( \tan \theta \) and \( \sec \theta \) given that \( \sin \theta = \frac{5}{13} \).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We know that \( \sin \theta = \frac{5}{13} \). In a right triangle, this means:
- The length of the side opposite to angle \( \theta \) (perpendicular) is 5.
- The length of the hypotenuse is 13.

2. **Finding the Length of the Base**:
We can use the Pythagorean theorem to find the length of the base (adjacent side). The Pythagorean theorem states:
\[ AC^2 = AB^2 + BC^2 \]
Here, \( AC \) is the hypotenuse, \( AB \) is the opposite side (perpendicular), and \( BC \) is the adjacent side (base).
Substituting the known values:
\[ 13^2 = 5^2 + BC^2 \]
\[ 169 = 25 + BC^2 \]
Rearranging gives:
\[ BC^2 = 169 - 25 = 144 \]
Taking the square root:
\[ BC = \sqrt{144} = 12 \]

3. **Calculating \( \tan \theta \)**:
The tangent of an angle is defined as the ratio of the opposite side to the adjacent side:
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} = \frac{5}{12} \]

4. **Calculating \( \sec \theta \)**:
The secant of an angle is defined as the ratio of the hypotenuse to the adjacent side:
\[ \sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{AC}{BC} = \frac{13}{12} \]

5. **Final Values**:
Therefore, the values of \( \tan \theta \) and \( \sec \theta \) are:
\[ \tan \theta = \frac{5}{12}, \quad \sec \theta = \frac{13}{12} \]

### Summary:
The values of \( \tan \theta \) and \( \sec \theta \) respectively are \( \frac{5}{12} \) and \( \frac{13}{12} \).

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