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Jorden Pace 2022-07-03 Answered
Find all (x,y) satisfying ( sin 2 x + 1 sin 2 x ) 2 + ( cos 2 x + 1 cos 2 x ) 2 = 12 + 1 2 sin y
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Answers (1)

Jamarcus Shields
Answered 2022-07-04 Author has 17 answers
Using the inequality a 2 + b 2 ( a + b ) 2 2 , note that
( sin 2 x + 1 sin 2 x ) 2 + ( cos 2 x + 1 cos 2 x ) 2 1 2 ( sin 2 x + 1 sin 2 x + cos 2 x + 1 cos 2 x ) 2 = 1 2 ( 1 + cos 2 x + sin 2 x sin 2 x cos 2 x ) 2 = 1 2 ( 1 + 4 ( 2 sin x cos x ) 2 ) 2 = 1 2 ( 1 + 4 sin 2 2 x ) 2 1 2 ( 1 + 4 1 ) 2 = 25 2 .

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