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Simplify $\cos (\arcsin (\frac{1}{3})) = ?$

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\frac{2 \sqrt{2}}{5}

\frac{2 \sqrt{2}}{7}

\frac{2 \sqrt{2}}{9}

\frac{2 \sqrt{2}}{3}

answer is D.

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Detailed Solution

Here given equation

\cos (\arcsin (\frac{1}{3}))

Now, we must provide any variable an inverse sin value.

θ

will serve as the variable. So, let's take

\Rightarrow θ = \arcsin (\frac{1}{3})

_______(1)
This allows us to write

\Rightarrow sin θ = (\frac{1}{3})

We can determine the value of

cos θ

by using

sin θ

.
We know the fundamental trigonometric identity.

\sin^{2} θ + c {os}^{2} = 1

This allows us to write

cos θ = \sqrt{1 - \sin^{2} θ}

We can determine the

cos θ

using this

\Rightarrow cos θ = \sqrt{1 - {(\frac{1}{3})}^{2}}

\Rightarrow cos θ = \sqrt{1 - \frac{1}{9}}

\Rightarrow cos θ = \sqrt{\frac{8}{9}}

\Rightarrow cos θ = \frac{2 \sqrt{2}}{3}

We now get the value of

θ

in equation 1. Therefore, to obtain the answer, we must replace that

θ

with the value given.

θ = \arcsin (\frac{1}{3})

\Rightarrow \cos (\arcsin (\frac{1}{3}))

\Rightarrow \cos (\arcsin (\frac{1}{3})) = \frac{2 \sqrt{2}}{3}

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