Evaluate the following : $\dfrac{3\sin 30° - 4\sin^3 30°}{2\sin^2 50° + 2\cos^2 50°}$
Generated by claude-sonnet-4-6 · 2026-06-15 10:25 · grounding rag
Model Answer
Numerator: $3\sin 30° - 4\sin^3 30° = 3\times\dfrac{1}{2} - 4\times\left(\dfrac{1}{2}\right)^3 = \dfrac{3}{2} - 4\times\dfrac{1}{8} = \dfrac{3}{2} - \dfrac{1}{2} = 1$
Denominator: $2\sin^2 50° + 2\cos^2 50° = 2(\sin^2 50° + \cos^2 50°) = 2\times 1 = 2$
$$\therefore \quad \frac{3\sin 30° - 4\sin^3 30°}{2\sin^2 50° + 2\cos^2 50°} = \frac{1}{2}$$
Source: Chapter 8, Section 8.3 (Trigonometric Ratios of Specific Angles)
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Explanation
- Numerator uses $\sin 30° = \frac{1}{2}$ from the standard table.
- Denominator uses the identity $\sin^2\theta + \cos^2\theta = 1$, so $2(\sin^2 50° + \cos^2 50°) = 2$.
- Examiners award 1 mark each for correctly simplifying numerator and denominator. Show substitution steps clearly — don't skip directly to the answer.