What is the exact value of tan 30° - √3/2 when simplified?

Introduction / Context:
This trigonometry question asks for the exact value of an expression involving tan 30 degrees and a square root term. It tests knowledge of standard trigonometric values for special angles and the ability to combine them into a single simplified expression without converting to decimals.

Given Data / Assumptions:

• We need to evaluate tan 30° - √3/2.
• Standard exact values for trigonometric ratios at special angles can be used.
• The final answer should remain in exact fractional surd form.

Concept / Approach:
The key is to recall the exact value of tan 30°. Then we subtract √3/2 and simplify the resulting surd expression. When subtracting radicals, we must make sure they have the same basic root so that we can combine them like like terms. We also use simple fraction operations to combine the terms into a single denominator if needed.

Step-by-Step Solution:
Step 1: Recall that tan 30° = 1/√3. Step 2: Substitute this value into the expression: tan 30° - √3/2 becomes 1/√3 - √3/2. Step 3: To combine the terms, use a common denominator of 2√3. Step 4: Rewrite 1/√3 as 2/(2√3) and √3/2 as (3)/(2√3). Step 5: Now the expression becomes (2/(2√3)) - (3/(2√3)) = (2 - 3)/(2√3). Step 6: Simplify the numerator: 2 - 3 = -1, so the result is -1/(2√3).

Verification / Alternative check:
We can estimate numerically to check. tan 30° is approximately 0.577 and √3/2 is about 0.866. Their difference is about 0.577 - 0.866 = -0.289. The value -1/(2√3) is approximately -1/(2 * 1.732) which is -1/3.464, around -0.289. The close match confirms our exact simplification.

Why Other Options Are Wrong:

• Option B: (2√3 - √6)/(2√2) simplifies to a completely different value.
• Option C: (2 - √3)/2 is not equal to -1/(2√3) after rationalization.
• Option D: 1/6 is positive, while the true result is negative.
• Option E: 0 would require the two terms to be equal, which they are not.

Common Pitfalls:
Many learners confuse tan 30° with sin 30° and may incorrectly use 1/2 instead of 1/√3. Another mistake is to subtract the coefficients of the radicals directly without bringing them to a common denominator. Always express both terms with the same denominator or same radical base before combining them.

Final Answer:
Thus, the value of the expression is -1/(2√3).