sin(12cos−145)=

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Question

s i n (\frac{1}{2} c o s^{- 1} \frac{4}{5}) =

Options:

A . 1√10

B . −1√10

C . 110

D . −110

Answer: Option A
:
A
Let

0 \leq x \leq π

So,

c o s^{- 1} \frac{4}{5} = x \Rightarrow c o s x = \frac{4}{5}

....(i)
Now

s i n (\frac{1}{2} c o s^{- 1} \frac{4}{5}) = s i n (\frac{x}{2})

....(ii)
From (i),

c o s x = \frac{4}{5} \Rightarrow 1 - 2 s i n^{2} \frac{x}{2} = \frac{4}{5}

\Rightarrow 2 s i n^{2} \frac{x}{2} = 1 - \frac{4}{5} = \frac{1}{5} \Rightarrow s i n \frac{x}{2} = \sqrt{\frac{1}{10}}

.

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