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Geometry Difficulty 3.7 AMC 10/12 Find the answer Slovenia

What is the size of the angle xx in degrees, if 2cos⁡10∘+sin⁡100∘+sin⁡1000∘+sin⁡10000∘=sin⁡x2 \cos 10^\circ + \sin 100^\circ + \sin 1000^\circ + \sin 10000^\circ = \sin x and −90∘≤x≤90∘-90^\circ \le x \le 90^\circ holds?

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Solution

Since 2cos⁡10∘+sin⁡100∘+sin⁡1000∘+sin⁡10000∘=2cos⁡10∘+sin⁡80∘−sin⁡80∘−sin⁡80∘=sin⁡80∘2\cos 10^\circ + \sin 100^\circ + \sin 1000^\circ + \sin 10000^\circ = 2\cos 10^\circ + \sin 80^\circ - \sin 80^\circ - \sin 80^\circ = \sin 80^\circ, we have x=80∘x = 80^\circ.

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