`DeltaXYZ` is right angled at Y .If sin X `=(12)/(13)` ,then what is the value of cosec Z?

Options

A. a.`(5)/(12)`
B. b.`(5)/(13)`
C. c.`(13)/(5)`
D. d.`(13)/(12)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the triangle and given information
We have a right-angled triangle ΔXYZ, where the right angle is at Y. We are given that sin X = 12/13.

### Step 2: Identify the sides of the triangle
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse. Here, if we denote:
- XY as the opposite side to angle X (perpendicular),
- XZ as the hypotenuse,
- YZ as the adjacent side to angle X (base).

From the information given:
- sin X = opposite/hypotenuse = YZ/XZ = 12/13

This means:
- Opposite (YZ) = 12
- Hypotenuse (XZ) = 13

### Step 3: Use Pythagorean theorem to find the adjacent side
Using the Pythagorean theorem:
\[ XZ^2 = XY^2 + YZ^2 \]
Substituting the known values:
\[ 13^2 = XY^2 + 12^2 \]
\[ 169 = XY^2 + 144 \]
\[ XY^2 = 169 - 144 \]
\[ XY^2 = 25 \]
Taking the square root:
\[ XY = \sqrt{25} = 5 \]

### Step 4: Find cosec Z
Cosecant (cosec) is defined as the reciprocal of sine. Therefore:
\[ \text{cosec Z} = \frac{1}{\sin Z} \]
Since angle Z is the complementary angle to angle X in a right triangle:
\[ \sin Z = \cos X \]
Using the definition of cosine:
\[ \cos X = \frac{adjacent}{hypotenuse} = \frac{XY}{XZ} = \frac{5}{13} \]
Thus:
\[ \sin Z = \cos X = \frac{5}{13} \]
Now, we can find cosec Z:
\[ \text{cosec Z} = \frac{1}{\sin Z} = \frac{1}{\frac{5}{13}} = \frac{13}{5} \]

### Final Answer
The value of cosec Z is \( \frac{13}{5} \).

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