Day 7: Seven Terms in a Tricky Trig Sum

Geometry Level 5

Solve: 2 sin ⁡ θ ( sin ⁡ 2 θ + sin ⁡ 4 θ + sin ⁡ 6 θ + ⋯ + sin ⁡ 14 θ ) = cos ⁡ θ − 1 2 2\sin\theta(\sin2\theta +\sin4\theta +\sin6\theta +\dots+\sin14\theta ) = \cos\theta - \frac12 where 0 ∘ ≤ θ ≤ 2 4 ∘ 0^{\circ}\leq \theta \leq 24^{\circ} .

Give your answer as the product of your solutions (each in degrees).

This problem is part of the Advent Calendar 2015 .


The answer is 80.

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Michael Ng by Michael Ng , 22, United Kingdom

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3 solutions

Michael Ng
Dec 6, 2015

I will type up a solution soon, but here is a written version (This is part of a challenging trigonometry sheet that I wrote).

And therefore the answer is 4 × 20 = 80 4 \times 20 =\boxed{80} , as required.

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Akhilesh Vibhute
Dec 17, 2015

2SinASinB = Cos(A-B)-Cos(A+B) Simplify you'll get cos15∆=1/2 so 4 and 20 is the solution. As simple as that.....

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Aakash Khandelwal
Dec 15, 2015

Use sin(a)+sin(a+d)....sin(a+(n-1)d)=sin(a+(n-1)d/2) sin(n d/2)/sin(d/2).

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Yes a nice identity indeed!

Michael Ng - 5 years, 6 months ago

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