NIMCET 2008 — Mathematics PYQ

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Question

NIMCET 2008 — Mathematics PYQ

NIMCET | Mathematics | 2008

The value of $\sin 12° \sin 48° \sin 54°$ is:

Choose the correct answer:

Explanation

1. Apply the Product-to-Sum Identity

We use the identity: $\sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)]$ .

Let $A = 48°$ and $B = 12°$ :

\sin 48° \sin 12° = \frac{1}{2} [\cos(48° - 12°) - \cos(48° + 12°)]

\sin 48° \sin 12° = \frac{1}{2} [\cos 36° - \cos 60°]

2. Substitute the Known Values

We know that $\cos 60° = 1/2$ . The expression becomes:

\frac{1}{2} \left[ \cos 36° - \frac{1}{2} \right]

Now multiply this by the remaining term from the question, $\sin 54°$ :

\text{Expression} = \frac{1}{2} \left[ \cos 36° - \frac{1}{2} \right] \sin 54°

3. Use Complementary Angle Relationship

Note that $\sin 54° = \cos(90° - 54°) = \cos 36°$ .

Substitute $\sin 54° = \cos 36°$ into the expression:

\text{Expression} = \frac{1}{2} \left[ \cos 36° - \frac{1}{2} \right] \cos 36°

\text{Expression} = \frac{1}{2} \cos^2 36° - \frac{1}{4} \cos 36°

4. Substitute the value of $\cos 36°$

The value of $\cos 36°$ is $\frac{\sqrt{5}+1}{4}$ .

\text{Expression} = \frac{1}{2} \left( \frac{\sqrt{5}+1}{4} \right)^2 - \frac{1}{4} \left( \frac{\sqrt{5}+1}{4} \right)

\text{Expression} = \frac{1}{2} \left( \frac{5 + 1 + 2\sqrt{5}}{16} \right) - \frac{\sqrt{5}+1}{16}

\text{Expression} = \frac{6 + 2\sqrt{5}}{32} - \frac{2\sqrt{5} + 2}{32}

\text{Expression} = \frac{6 + 2\sqrt{5} - 2\sqrt{5} - 2}{32}

\text{Expression} = \frac{4}{32} = \frac{1}{8}

5. Compare with Options

We know that $\sin 30° = 1/2$ .

(a) $\sin 30° = 1/2$
(b) $\sin^2 30° = (1/2)^2 = 1/4$
(c) $\sin^3 30° = (1/2)^3 = 1/8$

The calculated value $1/8$ matches $\sin^3 30°$ .

Correct Option: (c) $\sin^3 30°$