Using standard trigonometric values, what is the exact value of Sin 60° + (1/2) Cosec 45°?

Introduction / Context:
This problem checks knowledge of standard trigonometric values, especially for special angles 45 degrees and 60 degrees. Such values are widely used in geometry, physics, and aptitude tests. The question combines sine and cosecant functions and requires correct substitution followed by simple fraction addition.

Given Data / Assumptions:

• We need to evaluate Sin 60° + (1/2) Cosec 45°.
• Angles are in degrees.
• We use exact special angle values, not decimal approximations.

Concept / Approach:
The key is to recall that Sin 60° = √3 / 2 and Sin 45° = √2 / 2. Cosecant is the reciprocal of sine, so Cosec 45° = 1 / Sin 45°. After substituting these values, we simplify the expression by putting both terms over a common denominator. The goal is to match the simplified result with one of the provided options.

Step-by-Step Solution:
Recall Sin 60° = √3 / 2. Recall Sin 45° = √2 / 2, so Cosec 45° = 1 / Sin 45° = 2 / √2 = √2. Compute (1/2) Cosec 45° = (1/2) * √2 = √2 / 2. Now add the two terms: Sin 60° + (1/2) Cosec 45° = √3 / 2 + √2 / 2. Combine over the common denominator 2 to get (√3 + √2) / 2. This matches option (√3 + √2)/2 exactly.

Verification / Alternative check:
If we approximate numerically, Sin 60° is about 0.866, and Cosec 45° is the reciprocal of about 0.707, which is approximately 1.414. Then (1/2) Cosec 45° is about 0.707. Adding gives approximately 1.573. Evaluating (√3 + √2)/2 with √3 ≈ 1.732 and √2 ≈ 1.414 gives (1.732 + 1.414)/2 ≈ 3.146/2 ≈ 1.573, confirming the algebraic result.

Why Other Options Are Wrong:
(√3 + 2)/2 and (√3 + 2)/√2 arise if one mistakenly takes Cosec 45° as 2 instead of √2. The form (√3 + √2)/√2 has the wrong denominator, and (2√3 + √2)/4 does not simplify to the correct expression. None of these reflect the precise substitution of special angle values.

Common Pitfalls:
Students sometimes confuse sine with cosine values for 30 degrees, 45 degrees, and 60 degrees, or they invert the wrong ratio when finding cosecant. Another mistake is forgetting to simplify 2 / √2 to √2, leaving extra factors in the denominator. Keeping a small table of key exact values in memory is very helpful for such questions.

Final Answer:
The exact value of the expression is (√3 + √2)/2.