Evaluate the following: cosec-1(-2)+cot-1(3) - Mathematics and Statistics

प्रश्न

Evaluate the following:

${cosec}^{- 1} (- \sqrt{2}) + \cot^{- 1} (\sqrt{3})$

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उत्तर

Let ${cosec}^{- 1} (- \sqrt{2}) = α, where \frac{- π}{2} \leq y \leq \frac{π}{2}, y \neq 0$

∴ cosec α = $- \sqrt{2} = - cosec \frac{π}{4}$

∴ cosec α = $cosec (- \frac{π}{4})$ ...[∵ cosec (– θ) = – cosec θ]]

∴ α = $- \frac{π}{4} ... [∵ \frac{- π}{2} \leq \frac{- π}{4} \leq \frac{π}{2}]$

∴ ${cosec}^{- 1} (- \sqrt{2}) = - \frac{π}{4}$ ...(1)

Let $\cot^{- 1} (\sqrt{3})$ = β, where 0 < β < π

∴ cot β = $\sqrt{3} = \cot \frac{π}{6}$

∴ β = $\frac{π}{6} ... [∵ 0 < \frac{π}{6} < π]$

∴ $\cot^{- 1} (\sqrt{3}) = \frac{π}{6}$ ...(2)

∴ ${cosec}^{- 1} (- \sqrt{2}) + \cot^{- 1} (\sqrt{3})$

= $- \frac{π}{4} + \frac{π}{6}$ ...[By (1) and (2)]

= $\frac{- 3 π + 2 π}{12}$

= $- \frac{π}{12}$ .

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Inverse Trigonometric Functions

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Q 2.4 Q 2.3 Q 3.1

अध्याय 3: Trigonometric Functions - Exercise 3.3 [पृष्ठ १०३]

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अध्याय 3 Trigonometric Functions
Exercise 3.3 | Q 2.4 | पृष्ठ १०३

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Evaluate the following: cosec-1(-2)+cot-1(3) - Mathematics and Statistics

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Evaluate the following: cosec-1(-2)+cot-1(3) - Mathematics and Statistics

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