First slide

Properties of ITF

Question

$2 π - (\sin^{- 1} \frac{4}{5} + \sin^{- 1} \frac{5}{13} + \sin^{- 1} \frac{16}{65}) is equal to:$

Moderate

A

$\frac{π}{2}$
B

$\frac{7 π}{4}$
C

$\frac{3 π}{2}$
D

$\frac{5 π}{2}$

Solution

$\begin{array}{l} 2 π - [\tan^{- 1} \frac{4}{3} + \tan^{- 1} \frac{5}{12} + \tan^{- 1} \frac{16}{63}] \\ \Rightarrow 2 π - [\tan^{- 1} (\frac{\frac{4}{5} + \frac{5}{12}}{1 - \frac{4}{3} \cdot \frac{5}{12}}) + \tan^{- 1} \frac{16}{63}] \\ \Rightarrow 2 π - [\tan^{- 1} \frac{63}{16} + \cot^{- 1} \frac{63}{16}] = 2 π - \frac{π}{2} = \frac{3 π}{2} \end{array}$

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