Trignometry - 1
sin 9 π sin 9 2 π sin 9 4 π = b a
The equation above holds true for coprime integers a and b . Find ∣ a − b ∣ .
The answer is 5.
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1 solution
Is that result a standard-one? :o
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No, but you can derive it easily :)
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Can you please tell how has it been derived? \
Krishna Ar so please derive it although you are such a genius
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@Harshi Singh – Hey, use these results:
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sin ( A + B ) sin ( A − B ) = sin 2 A − sin 2 B
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sin ( 3 θ ) = 3 sin θ − 4 sin 3 θ
@Harshi Singh – We know that sin(A+B)sin(A-B)= sin^2A-sin^2B And Sin3A=3sinA-4sin^3A This implies that sinAsin(60+A)sin(60-A)=sinA(sin^2(60)-sin^2A) =sinA(3/4-sin^2A) =3/4sinA - sin^3A =1/4(3sinA-4sin^3A) =1/4sin3A
Its that simple
Use the following result: sin ( θ ) sin ( 3 π − θ ) sin ( 3 π + θ ) = 4 1 sin ( 3 θ ) In this context, θ = 9 π . And so our expression reduces to 4 1 sin ( 3 × 9 π ) = 4 1 sin ( 3 π ) = 8 3 And our answer is ∣ 3 − 8 ∣ = 5