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The maximum value of 12 sin(theta) - 9 sin ^2 (theta) is

a) 3 b) 4

c) 5 d) none of these

Let z = sin x

Then

y = 12 sin (x)- 9 sin^2 (x)

becomes

y = 12z - 9z^2 = 3z(4 - 3z)

which has roots at 0 and 4/3

The vertex is half-way between 0 and 4/3:

z = (0 + 4/3)/2 = 2/3

This is good because sin x = 2/3 has a

solution. We don't need to know what

it is, but we need to know that it exists.

The maximum value occurs at the vertex:

y_max = 3(2/3)(4 - 3(2/3)) = 2(4 - 2) = 4

Regards

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