`DeltaPQR` is right angled at Q .If cos `P=(5)/(13)`,then what is the value of cosec R?

Options

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

Answer

To solve the problem, we need to find the value of cosec R in the right-angled triangle PQR, where the right angle is at Q and cos P = 5/13.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
In triangle PQR, angle Q is the right angle. This means that angles P and R are complementary, i.e., P + R = 90°.

2. **Using the Cosine Definition**:
We know that cos P = adjacent side (PQ) / hypotenuse (PR). Given that cos P = 5/13, we can identify:
- Adjacent side (PQ) = 5
- Hypotenuse (PR) = 13

3. **Finding the Opposite Side (QR)**:
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 using the definition of sine:
\[ \sin R = \text{opposite side (QR)} / \text{hypotenuse (PR)} = QR / PR = 12 / 13 \]

5. **Finding cosec R**:
Cosec R is the reciprocal of sin R:
\[ \csc R = 1 / \sin R = 1 / (12/13) = 13/12 \]

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

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