In right-angled triangle ABC, right angled at B, if ∠A = 45° and hence ∠C = 45°, what is the exact simplified value of the expression (cosec C + 1/√3)?

Introduction / Context:This problem uses trigonometric values for a 45 degree angle and asks you to combine cosecant with a constant fraction involving sqrt(3). The triangle is right angled at B with the other two angles both equal to 45 degrees, making it a special isosceles right triangle. The purpose is to apply standard trigonometric values and then perform a careful algebraic simplification.

Given Data / Assumptions:

• Triangle ABC is right angled at B, so angle B = 90°.
• Angle A = 45°, so angle C is also 45°.
• We must compute the expression cosec C + 1/√3.
• Standard exact values for trigonometric ratios at 45° are used.

Concept / Approach:For an angle of 45 degrees in a right triangle, sin 45° is 1 / sqrt(2), so cosec 45° is its reciprocal, sqrt(2). Once we know cosec C, which is cosec 45°, we simply add 1/√3. To match the answer options, it is useful to combine these two terms into a single fraction and simplify the surds. This involves using a common denominator and possibly rationalising if needed.

Step-by-Step Solution:Because angle C = 45°, cosec C = cosec 45°.Recall sin 45° = 1 / sqrt(2), so cosec 45° = 1 / sin 45° = sqrt(2).The expression becomes sqrt(2) + 1/√3.Write both terms over a common denominator of √3: sqrt(2) = (sqrt(2) * √3) / √3 = sqrt(6) / √3.Thus sqrt(2) + 1/√3 = (sqrt(6) / √3) + (1 / √3) = (sqrt(6) + 1) / √3.Therefore, the simplified expression is (√6+1)/√3.

Verification / Alternative check:Use approximate decimal values. Take sqrt(2) approximately equal to 1.414 and sqrt(3) approximately equal to 1.732. Then 1/√3 is about 0.577. The original expression sqrt(2) + 1/√3 is about 1.414 + 0.577 = 1.991. Now evaluate (√6+1)/√3. Since sqrt(6) is about 2.449, the numerator is about 3.449. Dividing by 1.732 gives approximately 1.991, which matches, confirming the algebraic simplification.

Why Other Options Are Wrong:

• (3√2+1)/3 and (2+√3)/2 evaluate to different decimal values and do not equal about 1.991.
• 5/√3 equals roughly 5/1.732 ≈ 2.886, which is much larger.
• The value 2 is only an approximation and does not match the exact surd expression.
• Only (√6+1)/√3 matches both the exact and approximate value of the given expression.

Common Pitfalls:

• Using the wrong value for sin 45° or cosec 45°.
• Failing to find a common denominator when adding sqrt(2) and 1/√3.
• Making mistakes when multiplying surds, for example confusing sqrt(2) * sqrt(3) with sqrt(5) instead of sqrt(6).

Final Answer:(√6+1)/√3