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  • Provide solution for RD Sharma maths class 12 chapter Inverse Trignometric Functions exercise 3.7 question 5 sub question (iv)

Provide solution for RD Sharma maths class 12 chapter Inverse Trignometric Functions exercise  3.7 question 5 sub question (iv)

Answers (1)

Answer:    -\frac{\pi }{6}

Hint: The range of principal value of    \operatorname{cosec}^{-1} \text { is }\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\}
Given:  \operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{11 \pi}{6}\right)

Explanation:

First we solve  \operatorname{cosec}\left(\frac{11 \pi}{6}\right)

            \operatorname{cosec}\left(\frac{11 \pi}{6}\right)=\operatorname{cosec}\left(2 \pi-\frac{\pi}{6}\right)

As we know, \operatorname{cosec}(2 \pi-\theta)=-\operatorname{cosec}(\theta)

            \operatorname{cosec}\left(2 \pi-\frac{\pi}{6}\right)=-\operatorname{cosec}\left(\frac{\pi}{6}\right)

As we know,  \operatorname{cosec}\left(\frac{\pi}{6}\right)=2

            -\operatorname{cosec}\left(\frac{\pi}{6}\right)=-2

By substituting these values in \operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{11 \pi}{6}\right) we get,

              \operatorname{cosec}^{-1}(-2)

Let,        y=\operatorname{cosec}^{-1}(-2)

              \begin{aligned} &\operatorname{cosec} y=-2 \\ &-\operatorname{cosec} y=2 \\ &-\operatorname{cosec}\left(\frac{\pi}{6}\right)=2 \end{aligned}

As we know   \operatorname{cosec}(-\theta)=-\operatorname{cosec} \theta

               -\operatorname{cosec} \frac{\pi}{6}=\operatorname{cosec}\left(-\frac{\pi}{6}\right)

The range of principal value of    \operatorname{cosec}^{-1} \text { is }\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\} \text { and } \operatorname{cosec}\left(-\frac{\pi}{6}\right)=-2

            \therefore \operatorname{cosec}^{-1}\left(\operatorname{cosec}\left(\frac{11 \pi}{6}\right)\right) \mathrm{is}-\frac{\pi}{6}

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