Olympiad Maths Prep

Track / Stage 3 / 198 of 260 #198 of 2000

Problem 198

AMC 10/12, early questions
Geometry Difficulty 3.7 Find the answer National Math Olympiad · 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?

(A) -80
(B) -10
(C) 0
(D) 10
(E) 80

Official 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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