Evaluate: `Sin{Cos^-1(-3/5)+Cot^-1(-5/12)}` - Mathematics

प्रश्न

Evaluate: $\sin {\cos^{- 1} (- \frac{3}{5}) + \cot^{- 1} (- \frac{5}{12})}$

उत्तर

$\sin {\cos^{- 1} (- \frac{3}{5}) + \cot^{- 1} (- \frac{5}{12})} = \sin {π - \cos^{- 1} (\frac{3}{5}) + π - \cot^{- 1} (\frac{5}{12})}$

$= \sin {2 π - [\cos^{- 1} (\frac{3}{5}) + \cot^{- 1} (\frac{5}{12})]}$

$= - \sin {\cos^{- 1} (\frac{3}{5}) + \cot^{- 1} (\frac{5}{12})}$

$= - \sin {\sin^{- 1} [\sqrt{1 - {(\frac{3}{5})}^{2}}] + \sin^{- 1} [\frac{\frac{12}{5}}{\sqrt{1 + {(\frac{12}{5})}^{2}}}]}$

$= - \sin (\sin^{- 1} \frac{4}{5} + \sin^{- 1} \frac{12}{13})$

$= - \sin {\sin^{- 1} [\frac{4}{5} \times \sqrt{1 - {(\frac{12}{13})}^{2}} = \frac{12}{13} \times \sqrt{1 - {(\frac{4}{5})}^{2}}]}$

$= - \sin [\sin^{- 1} (\frac{20}{65} + \frac{36}{65})]$

$= - \sin [\sin^{- 1} (\frac{56}{65})]$

$= - \frac{56}{65}$

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Inverse Trigonometric Functions (Simplification and Examples)

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अध्याय 4: Inverse Trigonometric Functions - Exercise 4.09 [पृष्ठ ५९]

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अध्याय 4 Inverse Trigonometric Functions
Exercise 4.09 | Q 3 | पृष्ठ ५९

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