JEE Main 2026 Apr 2 Shift 1, Mathematics Q11: Trigonometric Ratios And Identities

JEE Main2026Apr 2, Shift 1Mathematics

If $sin (\frac{π}{18}) sin (\frac{5 π}{18}) sin (\frac{7 π}{18}) = K$ , then the value of $sin (\frac{10 K π}{3})$ is :

Solution

The three angles $π /18, 5 π /18, 7 π /18$ are $1 0^{\circ}, 5 0^{\circ}, 7 0^{\circ}$ . Use the identity

$sin θ sin (6 0^{\circ} - θ) sin (6 0^{\circ} + θ) = \frac{1}{4} sin 3 θ$

with $θ = 1 0^{\circ}$ (so the angles are $1 0^{\circ}, 5 0^{\circ}, 7 0^{\circ}$ ):

$K = \frac{1}{4} sin 3 0^{\circ} = \frac{1}{8}$

Then $\frac{10 K π}{3} = \frac{10 π}{24} = \frac{5 π}{12}$ , which equals $7 5^{\circ}$ .

$sin 7 5^{\circ} = sin (4 5^{\circ} + 3 0^{\circ}) = \frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} = \frac{3 + 1}{2 2}$ .

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