CoSine-ception

Geometry Level 3

∏ n = 0 9 cos ⁡ ( 2 ∘ ) n \large \prod_{n=0}^{9}\cos (2^\circ)^n

Evaluate the expression above in terms of the sine function. Express each angle as a positive angle less than 18 0 ∘ 180^\circ .

sin ⁡ 4 6 ∘ 1024 sin ⁡ 1 ∘ \frac{\sin46^\circ}{1024\sin1^\circ} − sin ⁡ 5 6 ∘ 1024 sin ⁡ 1 ∘ \frac { -\sin { 56^\circ } }{ 1024\sin { 1^\circ } } sin ⁡ 5 6 ∘ 1024 sin ⁡ 1 ∘ \frac{\sin56^\circ}{1024\sin1^\circ} − sin ⁡ 4 6 ∘ 1024 sin ⁡ 1 ∘ \frac{-\sin46^\circ}{1024\sin1^\circ}

Raph Pane by Raph Pane , 16, Philippines

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1 solution

∏ n = 0 9 cos ⁡ ( 2 ∘ ) n = cos ⁡ 1 ∘ cos ⁡ 2 ∘ cos ⁡ 4 ∘ cos ⁡ 8 ∘ ⋯ cos ⁡ 51 2 ∘ = sin ⁡ 1 ∘ cos ⁡ 1 ∘ cos ⁡ 2 ∘ cos ⁡ 4 ∘ cos ⁡ 8 ∘ ⋯ cos ⁡ 51 2 ∘ sin ⁡ 1 ∘ = sin ⁡ 2 ∘ cos ⁡ 2 ∘ cos ⁡ 4 ∘ cos ⁡ 8 ∘ ⋯ cos ⁡ 51 2 ∘ 2 sin ⁡ 1 ∘ = sin ⁡ 4 ∘ cos ⁡ 4 ∘ cos ⁡ 8 ∘ cos ⁡ 1 6 ∘ ⋯ cos ⁡ 51 2 ∘ 4 sin ⁡ 1 ∘ = sin ⁡ 8 ∘ cos ⁡ 8 ∘ cos ⁡ 1 6 ∘ cos ⁡ 3 2 ∘ ⋯ cos ⁡ 51 2 ∘ 8 sin ⁡ 1 ∘ = ⋯ ⋯ ⋯ = sin ⁡ 102 4 ∘ 1024 sin ⁡ 1 ∘ = sin ⁡ 30 4 ∘ 1024 sin ⁡ 1 ∘ = − sin ⁡ 5 6 ∘ 1024 sin ⁡ 1 ∘ \begin{aligned} \prod_{n=0}^9 \cos (2^\circ)^n & = \cos 1^\circ \cos 2^\circ \cos 4^\circ \cos 8^\circ \cdots \cos 512^\circ \\ & = \frac {\blue{\sin 1^\circ}\cos 1^\circ \cos 2^\circ \cos 4^\circ \cos 8^\circ \cdots \cos 512^\circ}{\blue{\sin 1^\circ}} \\ & = \frac {\blue{\sin 2^\circ}\cos 2^\circ \cos 4^\circ \cos 8^\circ \cdots \cos 512^\circ}{\blue 2 \sin 1^\circ} \\ & = \frac {\blue{\sin 4^\circ}\cos 4^\circ \cos 8^\circ \cos 16^\circ \cdots \cos 512^\circ}{\blue 4 \sin 1^\circ} \\ & = \frac {\blue{\sin 8^\circ}\cos 8^\circ \cos 16^\circ \cos 32^\circ \cdots \cos 512^\circ}{\blue 8 \sin 1^\circ} \\ & = \quad \cdots \quad \cdots \quad \cdots \\ & = \frac {\sin 1024^\circ}{1024 \sin 1^\circ} = \frac {\sin 304^\circ}{1024 \sin 1^\circ} = \boxed{\frac {- \sin 56^\circ}{1024 \sin 1^\circ}} \end{aligned}

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