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manierato5h

manierato5h

Answered question

2022-06-15

I have this inequality log tan x sin 2 x 5 12 < 1 that I would like to solve.

Answer & Explanation

Sage Mcdowell

Sage Mcdowell

Beginner2022-06-16Added 19 answers

1) If tan x > 1 π 4 < x < π 2 or 5 π 4 < x < 3 π 2 (1) then:
sin 2 x 5 12 < tan x sin 2 x 5 12 < tan 2 x sin 2 x tan 2 x < 5 12 sin 4 x cos 2 x < 5 12 12 sin 4 x < 5 ( 1 sin 2 x ) 12 sin 4 x 5 sin 2 x + 5 > 0
which is true for any x R (2).
We also must have
sin 2 x 5 12 sin x 15 6  or  sin x 15 6 ( 3 )
So, considering the intersection of (1),(2) and (3) we get the solution
π 4 < x < π 2  or  5 π 4 < x < 3 π 2

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