In right-triangle trigonometry, cosec X = 1/sin X relates an angle to the ratio of hypotenuse and opposite side. In triangle XYZ, angle Y = 90 degrees. If cosec X = 13/12 and XY = 1 cm, what is the length of side YZ (in cm)?

Introduction / Context:This problem tests how to convert a trigonometric ratio (cosec) into side ratios in a right-angled triangle and then scale those ratios using a given side length. It combines knowledge of cosec X, the definition of sine, and the idea of similar right-triangle side triples.

Given Data / Assumptions:

• Triangle XYZ is right-angled at Y (angle Y = 90 degrees).
• cosec X = 13/12, so sin X = 12/13.
• XY = 1 cm.
• For angle X: opposite side is YZ, adjacent side is XY, hypotenuse is XZ.

Concept / Approach:Use cosec X = hypotenuse/opposite. From cosec X = 13/12, we get XZ:YZ = 13:12. A right triangle with sides in the ratio 5:12:13 is a standard Pythagorean triple, meaning the adjacent side to angle X is 5 parts when hypotenuse is 13 parts and opposite is 12 parts.

Step-by-Step Solution: cosec X = 13/12 means XZ / YZ = 13/12 So let XZ = 13k and YZ = 12k for some scale factor k Because angle Y is 90 degrees, Pythagoras applies: XY^2 + YZ^2 = XZ^2 XY^2 = XZ^2 - YZ^2 = (13k)^2 - (12k)^2 = 169k^2 - 144k^2 = 25k^2 So XY = 5k Given XY = 1 cm, we get 5k = 1 => k = 0.2 Then YZ = 12k = 12 * 0.2 = 2.4 cm

Verification / Alternative check:Compute hypotenuse: XZ = 13k = 2.6 cm. Check: 1^2 + 2.4^2 = 1 + 5.76 = 6.76 and 2.6^2 = 6.76, so the triangle is consistent.

Why Other Options Are Wrong:2.0 and 1.8 are too small; they would not keep the 12:13 sin ratio with XY = 1.2.6 confuses YZ with the hypotenuse XZ.1.5 corresponds to a different side ratio and fails the Pythagoras check with hypotenuse 2.6.

Common Pitfalls:Mixing up which side is opposite angle X (it is YZ), or treating XY as the opposite side. Another common mistake is assuming cosec is opposite/hypotenuse instead of hypotenuse/opposite.

Final Answer:YZ = 2.4 cm