First slide

Properties of ITF

Question

Which of the following is/are a rational number?

Moderate

A

$\sin (\tan^{- 1} 3 + \tan^{- 1} \frac{1}{3})$
B

$\cos (\frac{π}{2} - \sin^{- 1} \frac{3}{4})$
C

$\log_{2} (\sin (\frac{1}{4} \sin^{- 1} \frac{\sqrt{63}}{8}))$
D

$\tan (\frac{1}{2} \cos^{- 1} \frac{\sqrt{5}}{3})$

Solution

$\begin{aligned} (1) & \sin (\tan^{- 1} 3 + \tan^{- 1} \frac{1}{3}) = \sin \frac{π}{2} = 1 \\ (2) & \cos (\frac{π}{2} - \sin^{- 1} \frac{3}{4}) = \cos (\cos^{- 1} \frac{3}{4}) = \frac{3}{4} \\ (3) & \sin (\frac{1}{4} \sin^{- 1} \frac{\sqrt{63}}{8}) \end{aligned}$
Let
$\begin{aligned} \sin^{- 1} \frac{\sqrt{63}}{8} = θ . Then, \\ \sin θ = \frac{\sqrt{63}}{8} \Rightarrow \cos θ = \frac{1}{8} \end{aligned}$
we have $\cos \frac{θ}{2} = \sqrt{\frac{1 + \cos θ}{2}} = \frac{3}{4}$
$\Rightarrow \sin \frac{θ}{4} = \sqrt{\frac{1 - \cos \frac{θ}{2}}{2}} = \frac{1}{2 \sqrt{2}}$
Now, $\log_{2} \sin (\frac{1}{4} \sin^{- 1} \frac{\sqrt{63}}{8}) = \log_{2} \frac{1}{2 \sqrt{2}} = - \frac{3}{2}$
(4)
$\cos^{- 1} \frac{\sqrt{5}}{3} = θ or \cos θ = \frac{\sqrt{5}}{3}$
$∴ \tan \frac{θ}{2} = \sqrt{\frac{1 - \cos θ}{1 + \cos θ}} = \frac{3 - \sqrt{5}}{2}$

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