In right angled triangle XYZ, which is right angled at Y, tan X = 15/8. What is the value of sin Z?

Introduction / Context:
This trigonometry question involves a right angled triangle and uses the relationships between the trigonometric ratios of complementary angles. It checks whether you can interpret tan X as a ratio of sides, construct the triangle, find the missing side using the Pythagoras theorem, and then compute sin Z for the other acute angle in the triangle.

Given Data / Assumptions:

• Triangle XYZ is right angled at Y.
• tan X = 15/8.
• Required: sin Z.
• Angles X and Z are acute and complementary since Y is 90 degrees.

Concept / Approach:
In a right angled triangle, for angle X, tan X is defined as (opposite side to X)/(adjacent side to X). Using tan X = 15/8, we can assign lengths to the sides opposite and adjacent to angle X in that ratio. Then we find the hypotenuse using the Pythagoras theorem. Finally, we compute sin Z = (opposite side to angle Z)/(hypotenuse). Since angles X and Z are complementary, we can also use the relation sin Z = cos X.

Step-by-Step Solution:
Step 1: For angle X in triangle XYZ, the side opposite X is YZ and the side adjacent to X is XY because the right angle is at Y. Step 2: Given tan X = 15/8, set YZ = 15k and XY = 8k for some positive constant k. Step 3: Use the Pythagoras theorem to find the hypotenuse XZ: XZ^2 = XY^2 + YZ^2 = (8k)^2 + (15k)^2 = 64k^2 + 225k^2 = 289k^2. Step 4: Take the square root to get XZ = sqrt(289k^2) = 17k. Step 5: For angle Z, the side opposite Z is XY and the hypotenuse is XZ. Step 6: Therefore sin Z = opposite/hypotenuse = XY/XZ = 8k/17k = 8/17. Step 7: This matches the idea that sin Z = cos X, and cos X = adjacent/hypotenuse = 8/17.

Verification / Alternative check:
You can verify by computing tan X using the constructed sides: tan X = YZ/XY = 15k/8k = 15/8, which agrees with the given value. Then confirm that sin Z = 8/17, and check that 8^2 + 15^2 = 17^2, which holds true. This ensures that the triangle is consistent and the calculated trigonometric values are correct.

Why Other Options Are Wrong:
15/17 corresponds to sin X, not sin Z. 17/8 and 17/15 are reciprocals of the tangent or cosine values and do not represent a valid sine ratio for an acute angle, since sine must be less than or equal to 1. The value 4/5 does not relate to the 8-15-17 Pythagorean triple used here and would imply inconsistent side lengths.

Common Pitfalls:
A common mistake is to assign the 15 and 8 to the wrong sides relative to angle X, which leads to incorrect side relationships. Another error is to forget that X and Z are complementary in a right triangle and that sin Z equals cos X. Carefully drawing the triangle and labelling sides according to the given ratio prevents these issues.

Final Answer:
The value of sin Z is 8/17.