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Question
If k = `sin(pi/18) sin((5pi)/18) sin((7pi)/18)`, then the numerical value of k is ______.
Solution
If k = `sin(pi/18) sin((5pi)/18) sin((7pi)/18)`, then the numerical value of k is `underlinebb(1/8)`.
Explanation:
Given that: k = `sin(pi/18) sin((5pi)/18) sin((7pi)/18)`
⇒ k = sin10°. sin50°. sin70°
⇒ k = sin10° sin(90° – 40°) sin(90° – 20°)
⇒ k = sin10° cos40° cos20°
⇒ k = `sin10^circ . 1/2 [2 cos 40^circ cos 20^circ]`
⇒ k = `sin 10^circ . 1/2 [cos(40^circ + 20^circ) + cos(40^circ - 20^circ)]`
⇒ k = `1/2 sin 10^circ [cos 60^circ + cos 20^circ]`
⇒ k = `1/2 sin 10^circ(1/2 + cos 20^circ)`
⇒ k = `1/4 sin 10^circ + 1/2 sin 10^circ . cos 20^circ`
⇒ k = `1/4 sin 10^circ + 1/4(2 sin 10^circ cos 20^circ)`
⇒ k = `1/4 sin 10^circ + 1/4[sin(10^circ + 20^circ) + sin(10^circ - 20^circ)]`
⇒ k = `1/4 sin 10^circ + 1/4[sin30^circ + sin(-10^circ)]`
⇒ k = `1/4 sin 10^circ + 1/4 sin 30^circ - 1/4 sin 10^circ`
= `1/4 sin 30^circ`
= `1/4 xx 1/2`
= `1/8`
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