In right angled triangle XYZ, angle Y is 90 degrees. If sin X = 4 / 5 and side XY is 6 centimetres, find the length of side YZ in centimetres.

Introduction / Context: This trigonometry question involves a right angled triangle and the sine of one of its acute angles. It shows how trigonometric ratios relate the sides of a right triangle and how we can use a known side and a given ratio to find another side.

Given Data / Assumptions:

• Triangle XYZ is right angled at Y, so angle Y = 90 degrees.
• sin X = 4 / 5.
• Side XY = 6 cm.
• We must find the length of side YZ in centimetres.

Concept / Approach: In a right triangle, sine of an angle is defined as: sin X = opposite side to angle X / hypotenuse. Here, with angle Y = 90 degrees, the hypotenuse is side XZ. For angle X, side opposite is YZ and side adjacent is XY. Given sin X = 4 / 5, the ratio of YZ to XZ is 4 : 5. We can scale this ratio using the given XY, which must match the adjacent side ratio corresponding to cos X.

Step-by-Step Solution: Step 1: For angle X, opposite side = YZ and hypotenuse = XZ. So sin X = YZ / XZ = 4 / 5. Step 2: This tells us the sides are in the ratio YZ : XZ = 4 : 5. Step 3: In a 3, 4, 5 type right triangle, if sin X = 4 / 5, then cos X = 3 / 5 and the sides follow XY : YZ : XZ = 3 : 4 : 5. Step 4: XY is adjacent to angle X, so XY corresponds to the ratio 3 parts. Step 5: Given XY = 6 cm, so 3 parts = 6 cm, which means 1 part = 6 / 3 = 2 cm. Step 6: YZ corresponds to 4 parts, so YZ = 4 * 2 = 8 cm. Step 7: Hypotenuse XZ would be 5 parts = 5 * 2 = 10 cm, which is consistent with the ratios.

Verification / Alternative check: We can recheck sin X using the computed sides. sin X = opposite / hypotenuse = YZ / XZ = 8 / 10 = 4 / 5, which matches the given value. Also, XY / XZ = 6 / 10 = 3 / 5, so cos X = 3 / 5, aligning perfectly with a 3, 4, 5 triangle pattern.

Why Other Options Are Wrong: 4 cm and 5 cm: These are smaller than the correct length and do not satisfy sin X = 4 / 5 with the given XY value. 10 cm and 12 cm: 10 cm is the hypotenuse, not YZ, and 12 cm would give sin X = 12 / 10 > 1, which is impossible.

Common Pitfalls: Students sometimes misidentify which side is opposite and which is adjacent, or they apply the ratio to the wrong side. Another common error is to treat 4 and 5 as the actual lengths rather than as parts of a ratio. Always map the ratio correctly to the sides, then scale using the known side length.

Final Answer: The length of side YZ is 8 cm.