If 12 -9 ^2x attains its maximum value at x= , then write the val… - Vidyadip

Question

If $12\sin\text{x}-9\sin^2\text{x}$ attains its maximum value at $\text{x}=\alpha,$ then write the value of $\sin\alpha.$

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Answer

$12\sin\text{x}-9\sin^2\text{x}$ has maximum value as 4, so

$12\sin\alpha-9\sin^2\alpha=4$

$9\sin^2\alpha-612\sin\alpha+4=0$

$9\sin^2\alpha-6\sin\alpha-6\sin\alpha+4=0$

$3\sin\alpha(3\sin\alpha-2)-2(3\sin\alpha-2)=0$

$(3\sin\alpha-2)(3\sin\alpha-2)=0$

$3\sin\alpha-2=0$

$\sin\alpha=\frac23$

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