Assume that 3.13 leq pi leq 3.15. The integer | vedclass

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(A) We are given the expression $\sin ^{-1}(\sin 1 \cos 4+\cos 1 \sin 4)$.Using the trigonometric identity $\sin(A+B) = \sin A \cos B + \cos A \sin B$

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$-1$

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(A) We are given the expression $\sin ^{-1}(\sin 1 \cos 4+\cos 1 \sin 4)$.Using the trigonometric identity $\sin(A+B) = \sin A \cos B + \cos A \sin B$,we can simplify the expression inside the inverse sine function:$\sin 1 \cos 4 + \cos 1 \sin 4 = \sin(1+4) = \sin 5$.So,the expression becomes $\sin ^{-1}(\sin 5)$.We know that $\sin ^{-1}(\sin x) = x$ only when $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.Since $3.13 \leq \pi \leq 3.15$,we have $1.565 \leq \frac{\pi}{2} \leq 1.575$.Since $5$ is not in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$,we find an integer $k$ such that $5 - 2k\pi \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.For $k=1$,$5 - 2\pi \approx 5 - 2(3.14) = 5 - 6.28 = -1.28$.Since $-1.575 \leq -1.28 \leq -1.565$,the value is $5 - 2\pi$.Given $\pi \approx 3.14$,the value is $5 - 6.28 = -1.28$.The integer closest to $-1.28$ is $-1$.

Assume that $3.13 \leq \pi \leq 3.15$. The integer closest to the value of $\sin ^{-1}(\sin 1 \cos 4+\cos 1 \sin 4)$,where $1$ and $4$ appearing in $\sin$ and $\cos$ are given in radians,is

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