Evaluate $\frac{2\sin 30 + 5\tan 60}{\sin 60}$, leaving your answer in surd form.

Analyze the question: We need the exact values of sin 30°, tan 60° and sin 60°, then simplify the fraction.

Concept: Exact trig values (from the 30-60-90 triangle):

• sin 30° = 1/2
• tan 60° = √3
• sin 60° = √3/2

Solution:

Numerator: 2 sin 30° + 5 tan 60° = 2(1/2) + 5√3 = 1 + 5√3

Denominator: sin 60° = √3/2

So the expression is:

(1 + 5√3) ÷ (√3/2) = 2(1 + 5√3)/√3 = (2 + 10√3)/√3

Split into two fractions:

2/√3 + 10√3/√3 = 2/√3 + 10

Rationalise 2/√3: 2/√3 × √3/√3 = 2√3/3

Result: 2√3/3 + 10

Check against the options: This matches option A exactly.

Answer: A

Exam tip: Memorise the table of exact values for 30°, 45° and 60°. When you divide by a fraction, multiply by its reciprocal. Then split the numerator so each term simplifies on its own, which is quicker than expanding everything. Always rationalise any surd left in the denominator, because JAMB options are written in rationalised form.