In right triangle DEF, right angled at E, if cot D = 5/12, what is the value of sin F?

Introduction / Context:
This question combines basic trigonometric ratios with properties of right triangles. We are given cot D in a right triangle and asked to find sin F, where F is another acute angle of the same triangle. The problem tests understanding that angles in a right triangle are related and that trigonometric ratios for different angles can be linked through complementary angle relationships.

Given Data / Assumptions:

• Triangle DEF is right angled at E.
• cot D = 5/12.
• We must find sin F.
• All sides of the triangle are positive, and angles D and F are acute.

Concept / Approach:
In a right triangle, the sum of the three angles is 180 degrees. Since angle E is 90 degrees, the remaining two angles D and F are complementary, so D + F = 90 degrees. That means F = 90 degrees - D. We also know that sin(90 degrees - D) = cos D. Therefore, sin F equals cos D. From cot D we can construct the sides of a reference triangle, then compute cos D as the ratio of the adjacent side to the hypotenuse.

Step-by-Step Solution:
Step 1: Use the relation for a right triangle: D + E + F = 180 degrees and E = 90 degrees. Step 2: Therefore D + F = 90 degrees, so F = 90 degrees - D. Step 3: Use the identity sin(90 degrees - D) = cos D, hence sin F = cos D. Step 4: Given cot D = 5/12, write cot D as adjacent / opposite = 5 / 12. Step 5: Consider a right triangle where side adjacent to D is 5k and side opposite D is 12k. The hypotenuse is then √(5k^2 + 12k^2) = √(25k^2 + 144k^2) = √(169k^2) = 13k. Step 6: cos D is adjacent / hypotenuse = 5k / 13k = 5/13. Step 7: Hence sin F = cos D = 5/13.

Verification / Alternative check:
We can also express sin F directly in terms of sides opposite F and the hypotenuse. Since F is the angle opposite the side that was adjacent to D, that side has length 5k, and the hypotenuse remains 13k. So sin F = 5k / 13k = 5/13 again, confirming the result through a second viewpoint.

Why Other Options Are Wrong:

• Option A: 5/12 is the ratio adjacent / opposite for angle D, not sin F.
• Option B: 13/5 is greater than 1 and cannot be a sine or cosine value.
• Option D: 13/12 is also greater than 1, so it is impossible for sin F.
• Option E: 12/13 equals sin D, not sin F.

Common Pitfalls:
A common mistake is to confuse the roles of D and F and forget that they are complementary. Another frequent error is to misinterpret cot D and try to use 12/5 instead of 5/12 or to compute the hypotenuse incorrectly. Drawing a small right triangle diagram with labelled sides usually helps keep the relationships clear.

Final Answer:
Therefore, the required value is sin F = 5/13.