Trigonometry! #150

Geometry Level 4

Find the greatest value of a a such that the equation cos ⁡ 2 x + a sin ⁡ x = 2 a − 7 \cos 2x+a\sin x=2a-7 possesses a solution.

This problem is part of the set Trigonometry .


The answer is 6.

Omkar Kulkarni by Omkar Kulkarni , 22, India

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2 solutions

Rohit Ner
Feb 25, 2015

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Mas Mus
May 6, 2015

cos ⁡ 2 x + a sin ⁡ x = 2 a − 7 − 2 sin ⁡ 2 x + a sin ⁡ x + 8 − 2 a = 0 \cos2x+a\sin x=2a-7\\-2\sin^2 x+a\sin x+8-2a=0

Note that − 1 ≤ sin ⁡ x ≤ 1 -1\leq\sin x\leq1 , so for sin ⁡ x = − 1 \sin x=-1 we have a = 2 a=2 and for sin ⁡ x = 1 \sin x=1 we have a = 6 a=6 . Hence, the greatest value of a a is 6 6

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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