Triangle ABC is right-angled at B. If ∠A = 30°, then ∠C is determined uniquely by angle sum. Find the exact value of sec C (do not approximate).

Introduction / Context: This problem tests basic angle relationships in a right triangle and the ability to compute a standard trigonometric ratio at a special angle. If a triangle is right-angled at one vertex, the other two acute angles must add up to 90°. Once the missing angle is found, secant can be computed using known values of cosine for special angles like 30° and 60°.

Given Data / Assumptions:

• Triangle ABC is right-angled at B (∠B = 90°) • ∠A = 30° • Required: sec C

Concept / Approach: Use the angle sum property of triangles: ∠A + ∠B + ∠C = 180°. Since ∠B = 90°, the acute angles satisfy: ∠A + ∠C = 90°. So ∠C = 90° - ∠A. Then compute sec C: sec C = 1 / cos C. For C = 60°, cos 60° is a standard value equal to 1/2, so sec 60° = 2.

Step-by-Step Solution: 1) Use triangle angle sum: ∠A + ∠B + ∠C = 180° 2) Substitute ∠B = 90° and ∠A = 30°: 30° + 90° + ∠C = 180° 3) Solve for ∠C: ∠C = 180° - 120° = 60° 4) Compute sec C: sec 60° = 1 / cos 60° 5) Use cos 60° = 1/2: sec 60° = 1 / (1/2) = 2

Verification / Alternative check: In a 30°-60°-90° triangle, the side ratios are 1 : √3 : 2 (opposite 30°, opposite 60°, hypotenuse). For angle 60°, cos 60° = adjacent/hypotenuse = 1/2, so sec 60° = 2. This matches the computed result from angle reasoning.

Why Other Options Are Wrong: • 1/2 is cos 60°, not sec 60°. • 1/√2 corresponds to sec 45°. • 1/√3 corresponds to cos 30° or cot 60°, not sec 60°. • √3 is closer to sec 30°, not sec 60°.

Common Pitfalls: • Forgetting that A and C add to 90° in a right triangle. • Mixing up sec and cos (sec is reciprocal of cos).

Final Answer: 2