`Delta`PQR is right angled at Q. If secP = 13/5, then what is the value of sinR ?

Options

A. `5//13`
B. `5//12`
C. `13//5`
D. `13//12`

Answer

To solve the problem, we need to find the value of sin R in triangle PQR, which is right-angled at Q, given that sec P = 13/5.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
- We have triangle PQR, which is right-angled at Q. This means that angle Q is 90 degrees.
- Therefore, angles P and R are complementary, meaning P + R = 90 degrees.

2. **Using the Secant Function**:
- We are given that sec P = 13/5.
- Recall that sec P is defined as the ratio of the length of the hypotenuse to the length of the adjacent side in triangle PQR.
- Therefore, if we let the hypotenuse (PR) = 13 and the adjacent side (PQ) = 5, we can denote the sides as follows:
- Hypotenuse (PR) = 13
- Adjacent side (PQ) = 5

3. **Finding the Opposite Side**:
- To find the length of the opposite side (QR), we can use the Pythagorean theorem:
\[ PR^2 = PQ^2 + QR^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + QR^2 \]
\[ 169 = 25 + QR^2 \]
\[ QR^2 = 169 - 25 = 144 \]
\[ QR = \sqrt{144} = 12 \]

4. **Finding sin R**:
- Now, we can find sin R. By definition, sin R is the ratio of the length of the opposite side (QR) to the length of the hypotenuse (PR):
\[ \sin R = \frac{QR}{PR} = \frac{12}{13} \]

5. **Conclusion**:
- Therefore, the value of sin R is \( \frac{12}{13} \).

### Final Answer:
The value of sin R is \( \frac{12}{13} \).