If sin (π/18) . sin (5π/18) . sin (7π/18) = K, then the value of sin (10Kπ/3) is:

The correct answer is Option 1. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q11 – JEE-MAIN 2026

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Home Maths Trigonometric FunctionsQuestion #484

\sin (\frac{π}{18}) \cdot \sin (\frac{5 π}{18}) \cdot \sin (\frac{7 π}{18}) = K

, then the value of

\sin (\frac{10 K π}{3})

is:

Clue 1 of 3: Recognize the pattern

The angles $\frac{π}{18}$ , $\frac{5 π}{18}$ , and $\frac{7 π}{18}$ can be expressed in the form $θ$ , $60^{\circ} - θ$ , and $60^{\circ} + θ$ .

🧠 Full Solution Path

Step 1: Evaluate K using trigonometric identity✦ Active

The given product is $\sin (\frac{π}{18}) \sin (\frac{5 π}{18}) \sin (\frac{7 π}{18})$ . Let $θ = \frac{π}{18} = 10^{\circ}$ . Then the angles are $10^{\circ}$ , $50^{\circ}$ , $70^{\circ}$ . These can be written as $θ$ , $60^{\circ} - θ$ , $60^{\circ} + θ$ . We use the identity $\sin θ \sin (60^{\circ} - θ) \sin (60^{\circ} + θ) = \frac{1}{4} \sin (3 θ)$ .

K = \sin (10^{\circ}) \sin (50^{\circ}) \sin (70^{\circ}) = \frac{1}{4} \sin (3 \times 10^{\circ}) = \frac{1}{4} \sin (30^{\circ})

K = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}

Step 2: Substitute K into the expression to be found○ Expand

Now we need to find the value of $\sin (\frac{10 K π}{3})$ . Substitute $K = \frac{1}{8}$ into the expression.

\sin (\frac{10 K π}{3}) = \sin (\frac{10 \times \frac{1}{8} \times π}{3}) = \sin (\frac{10 π}{24}) = \sin (\frac{5 π}{12})

Step 3: Calculate the final sine value○ Expand

Convert $\frac{5 π}{12}$ to degrees and evaluate the sine value. $\frac{5 π}{12} = \frac{5 \times 180^{\circ}}{12} = 5 \times 15^{\circ} = 75^{\circ}$ . We can use the sum formula for sine.

\sin (75^{\circ}) = \sin (45^{\circ} + 30^{\circ}) = \sin (45^{\circ}) \cos (30^{\circ}) + \cos (45^{\circ}) \sin (30^{\circ})

= \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \times \frac{1}{2}

= \frac{\sqrt{3}}{2 \sqrt{2}} + \frac{1}{2 \sqrt{2}} = \frac{\sqrt{3} + 1}{2 \sqrt{2}}

This matches option 1.

💡 Teacher's Secret Hint

Remember common trigonometric values for angles like $30^{\circ}, 45^{\circ}, 60^{\circ}$ and how to combine them for angles like $75^{\circ}$ .

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Maths Question 11 – JEE-MAIN 2026

Clue 1 of 3: Recognize the pattern

🧠 Full Solution Path