If 12 sin x − 9 sin²x Attains Maximum Value at x = α, Find sin α
Question:
If
\[
12\sin x-9\sin^2x
\]
attains its maximum value at
\[
x=\alpha
\]
then write the value of
\[
\sin\alpha
\]
Solution
Let \[ \sin x=t \]
Then, \[ 12\sin x-9\sin^2x = 12t-9t^2 \]
\[ = -9\left(t-\frac23\right)^2+4 \]
Maximum value occurs when \[ \left(t-\frac23\right)^2=0 \]
\[ t=\frac23 \]
Since \[ t=\sin x \] and \[ x=\alpha \]
\[ \boxed{\sin\alpha=\frac23} \]