How do you find the exact value of $cos (arcsin (\frac{1}{3}))$ $cos(arcsin(1/3))$ ?

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Answer 1 · 2016-08-28T06:57:03

$\frac{2 \sqrt{2}}{3}$ $(2sqrt2)/3$

Let $a = a r c sin (\frac{1}{3}) \in Q 1$ $a = arc sin (1/3) in Q1$ ,

for the principal value, wherein cosine is positive.

The given expression is

$cos a = \sqrt{1 - {sin}^{2} a} = \sqrt{1 - \frac{1}{9}} = \frac{2 \sqrt{2}}{3}$ $cos a = sqrt (1-sin^2a)= sqrt(1-1/9)=(2sqrt2)/3$ .

If Q2 value of $a = 180^{o} - Q 1 a$ $a = 180^o - Q1 a$ is permitted, the answer will be

$\pm (2 \sqrt{2}) / 3$ $+-(2sqrt2)//3$ ,

using

$sin (180^{o} - a) = sin a$ $sin (180^o-a)=sin a$ but $cos (180^{o} - a) = - cos a$ $cos(180^o-a)=- cos a$ .

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