In right triangle ABC, angle B is 90° and angle A is 60°. Using standard trigonometric ratios, evaluate the expression (1/√3) · cosec C, where C is the remaining angle of the triangle.

Introduction / Context:
This question uses angle sum of a triangle along with basic trigonometric ratios for standard angles. It tests understanding of acute angles in a right triangle and the definition of cosecant as the reciprocal of sine.

Given Data / Assumptions:

• Triangle ABC is right angled at B
• m∠A = 60°
• Therefore angle C = 30° (since A + B + C = 180°)
• We need to evaluate (1/√3) · cosec C

Concept / Approach:
First determine angle C using the sum of angles in a triangle. Since C is 30°, we use the standard value sine 30° = 1/2. Cosecant is defined as 1 / sin, so cosec 30° = 2. Multiplying this by 1/√3 gives the required answer.

Step-by-Step Solution:
Sum of angles: A + B + C = 180° Given B = 90° and A = 60° So C = 180° − 90° − 60° = 30° sin 30° = 1/2 Therefore cosec 30° = 1 / sin 30° = 1 / (1/2) = 2 Expression is (1/√3) · cosec C = (1/√3) · 2 So value = 2/√3

Verification / Alternative check:
As a quick check, approximate √3 ≈ 1.732. Then 2/√3 ≈ 2/1.732 ≈ 1.155. If we compute numerically, cosec 30° = 2 exactly, and 2 multiplied by approximately 0.577 (which is 1/√3) gives about 1.155. This is consistent with our simplified form 2/√3.

Why Other Options Are Wrong:
2/3 is smaller than 1 and does not match the numerical check. √2/√3 and √2/3 involve √2 which never appears in the original expression. The value 1 would require cosec C = √3/2 which is not correct for 30°.

Common Pitfalls:
A frequent error is confusing 30° and 60° values, especially interchanging sin 30° = 1/2 and sin 60° = √3/2. Another mistake is treating cosec as cos inverse instead of 1/sin. Remember that in a right triangle, once two angles are known, the third is determined uniquely.

Final Answer:
Hence, the value of (1/√3) · cosec C is 2/√3.