Using exact trigonometric values, what is the simplified value of the expression Sin 30° − √2 Cos 30°?

Introduction / Context:
This trigonometric simplification question combines exact values for sine and cosine at 30 degrees with a radical multiplier. It tests both recall of special angle values and algebraic handling of square roots in fractions, a common theme in trigonometry simplification exercises.

Given Data / Assumptions:

• We need to simplify Sin 30° − √2 Cos 30°.
• Angles are measured in degrees.
• Standard special angle values are to be used, not approximations.

Concept / Approach:
Recall the exact values Sin 30° = 1/2 and Cos 30° = √3 / 2. Substitute these into the expression and then simplify carefully. The radical √2 multiplies Cos 30°, so we obtain a product of √2 and √3, which becomes √6. Finally, we combine the terms over a common denominator to match the given options.

Step-by-Step Solution:
Recall Sin 30° = 1/2. Recall Cos 30° = √3 / 2. Compute √2 Cos 30° = √2 * (√3 / 2) = √6 / 2. Now substitute into the expression: Sin 30° − √2 Cos 30° = 1/2 − √6 / 2. Both terms have denominator 2, so combine them: (1 − √6) / 2. Thus the simplified exact value is (1 − √6)/2.

Verification / Alternative check:
We can verify by approximate numerical evaluation. Sin 30° is 0.5. Cos 30° is approximately 0.866, and √2 is about 1.414. Then √2 Cos 30° ≈ 1.414 * 0.866 ≈ 1.225. So the expression is 0.5 − 1.225 ≈ −0.725. Now evaluate (1 − √6)/2. Since √6 is about 2.449, 1 − 2.449 ≈ −1.449, and dividing by 2 gives about −0.7245, which matches the earlier decimal value within rounding error.

Why Other Options Are Wrong:
Expressions with denominator √2 such as (1 − √6)/√2 or (3 − √6)/√2 have different magnitudes and do not match the simplified exact result. Options (3 − √6)/2 and (1 − √3)/√2 come from misusing the values of sine or cosine or from incorrectly combining terms. Only (1 − √6)/2 is consistent with the correct substitution and simplification.

Common Pitfalls:
One frequent error is using incorrect special angle values, such as confusing Sin 30° with Cos 30°. Another mistake is mishandling the product √2 * √3, sometimes incorrectly writing it as √5. Also, students may forget to take the common denominator properly and end up with mismatched numerators. Careful substitution and stepwise simplification prevent such issues.

Final Answer:
The simplified value of the expression is (1 − √6)/2.