Compute ∑ k = 0 180 cos 2 ⁡ 2 k ∘

Question

Compute       ∑          k      =      0              180            cos    2    ⁡  2      k    ∘

Keely Fernandez · Accepted Answer

Hint: Observe that for the first quadrant, we have

\begin{aligned} \cos^{2} 2^{\circ} + \cos^{2} 4^{\circ} + \dots + \cos^{2} 88^{\circ} + \cos^{2} 90^{\circ} \\ = \cos^{2} 2^{\circ} + \cos^{2} 4^{\circ} + \dots + \cos^{2} 44^{\circ} + \sin^{2} 44^{\circ} + \sin^{2} 42^{\circ} + \dots + \sin^{2} 2^{\circ} + 0 \\ = 22. \end{aligned}

sebadillab0 · Accepted Answer

S  =      ∑          r      =      0              180            cos    2    ⁡  2      r    ∘    2  S  =      ∑          r      =      0              180        (  1  +  cos  ⁡  4      r    ∘    )  =  181  +      ∑          r      =      0              180        cos  ⁡  4      r    ∘  Using   ∑  cos when angles are in arithmetic progression,      ∑          r      =      0              180        cos  ⁡  4      r    ∘    =                    cos        ⁡        (        2        ⋅                  180          ∘                )        sin        ⁡        (        2        ⋅                  181          ∘                )                    sin        ⁡                  2          ∘                      =  +  1

Compute <munderover> ∑ <mrow class="MJX-TeXAtom-ORD"> k

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