In a right-angled triangle ABC, the right angle is at B and m∠A = 60°. Evaluate the expression: (2*sec C) * (1/2*sin A) What is the exact value?

Introduction / Context:This question combines triangle angle relationships with special-angle trigonometric values. Because it is a right triangle, if one acute angle is 60°, the other acute angle must be 30°. The expression also simplifies because (2)*(1/2) cancels.

Given Data / Assumptions:

• Triangle ABC is right-angled at B (so ∠B = 90°).
• m∠A = 60°.
• Therefore ∠C = 30° (since 60° + 90° + ∠C = 180°).
• Expression: (2*sec C) * (1/2*sin A).

Concept / Approach:Simplify constants first: (2)*(1/2) = 1, so the expression becomes sec C * sin A. Then use special values: sec 30° and sin 60°.

Step-by-Step Solution:

Verification / Alternative check:Notice sec 30° and sin 60° are reciprocal-scaled values that cancel exactly because cos 30° = sin 60°. So sec 30° = 1/cos 30° = 1/sin 60°, giving sec 30° * sin 60° = 1 directly.

Why Other Options Are Wrong:

Common Pitfalls:Forgetting that the third angle is 30°, or mixing up sec with cosec. Always simplify constants first because it reduces mistakes.

Final Answer:1