Question

Assertion (A): If \(\sin A=\frac{8}{17}\) then \(\cot A=\frac{15}{8}\).

Reason (R): In a \(\triangle A B C\), right angled at B, we have \(\sin A=\frac{B C}{A C}\) and \(\cot A=\frac{A B}{B C}\). 

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (А).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. Let the perpendicular and hypotenuse be 8k and 17k. By the Pythagorean theorem, the third side is \(\sqrt{(17k)^2-(8k)^2}=15k\). Hence, \(\cot A=\frac{15k}{8k}=\frac{15}{8}\).

Step 2 – Reason: R is true. In the stated right triangle, sine is the opposite side divided by the hypotenuse and cotangent is the adjacent side divided by the opposite side.

Step 3 – Link: R identifies the relevant side ratios; together with the Pythagorean theorem, these give the asserted cotangent. Thus, R correctly explains A.