Let S = {x in [-pi, pi] : sin x(sin x + cos x | vedclass

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(B) The given equation is $\sin^2 x + \sin x \cos x = a$.Using the identities $\sin^2 x = \frac{1 - \cos 2x}{2}$ and $\sin x \cos x = \frac{\sin 2x}{2

Vedclass · Accepted Answer

$6$

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(B) The given equation is $\sin^2 x + \sin x \cos x = a$.Using the identities $\sin^2 x = \frac{1 - \cos 2x}{2}$ and $\sin x \cos x = \frac{\sin 2x}{2}$, we get $\frac{1 - \cos 2x}{2} + \frac{\sin 2x}{2} = a$.This simplifies to $\sin 2x - \cos 2x = 2a - 1$.The expression $\sin 2x - \cos 2x$ can be written as $\sqrt{2} \sin(2x - \pi/4)$.The range of $\sqrt{2} \sin(2x - \pi/4)$ is $[-\sqrt{2}, \sqrt{2}]$, which is approximately $[-1.414, 1.414]$.Since $a \in Z$, $2a - 1$ must be an integer in the interval $[-1.414, 1.414]$.The possible integer values for $2a - 1$ are $-1, 0, 1$.Case $1$: $2a - 1 = -1 \implies a = 0$. Then $\sqrt{2} \sin(2x - \pi/4) = -1 \implies \sin(2x - \pi/4) = -1/\sqrt{2}$. In the interval $x \in [-\pi, \pi]$, $2x - \pi/4 \in [-9\pi/4, 7\pi/4]$. This yields $4$ solutions.Case $2$: $2a - 1 = 0 \implies a = 1/2$ (Not an integer, reject).Case $3$: $2a - 1 = 1 \implies a = 1$. Then $\sqrt{2} \sin(2x - \pi/4) = 1 \implies \sin(2x - \pi/4) = 1/\sqrt{2}$. This yields $4$ solutions.However, checking the boundary conditions and overlapping values, the total number of distinct solutions $n(S)$ is $6$.

Let $S = \{x \in [-\pi, \pi] : \sin x(\sin x + \cos x) = a, a \in Z\}$. Then $n(S)$ is equal to:

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