Trig Max Min Value

[sharp fjord]

How to find max-min value of this expression, without using calculator or graph?

[night kindle]

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[simple pilot]

Factorise 3sinx out and apply AM GM inequality

[sharp fjord]

so 3sinθ(4-3sinθ)
then (3sinθ)+(4-3sinθ)/2 ≥ √(3sinθ(4-3sinθ))
2 ≥ √(3sinθ(4-3sinθ))
4 ≥ 3sinθ(4-3sinθ) that's the maximum value right?

[simple pilot]

Yes

[sharp fjord]

so for the minimum value
-1 ≤ sinθ ≤ 1
-3 ≤ 3sinθ ≤ 3
1 ≤ 4-3sinθ ≤ 7
-21 ≤ 3sinθ(4-3sinθ) ≤ 21
so the minimum value is -21 correct?

or is there another way to compute minimum value