Question

Find the value of tan(sin^-1(1/2))

Answer 1

If you know that
`\sin\frac\pi3=\frac12`, then you know right away

`\tan\left(\sin^(-1)\frac12\right)=\tan\frac\pi3=\frac1{\sqrt}3=\frac{\sqrt3}3`

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Otherwise, you can derive the same result. Let
`\theta=\sin^(-1)\frac12`, so that
`\sin\theta=\frac12`.
`\sin^(-1)` is bounded, so we know
`-\frac\pi2\le\theta\le\frac\pi2`. For these values of
`\theta`, we always have
`\cos\theta\ge0`.

So, recalling the Pythagorean theorem, we find

`\cos^2\theta+\sin^2\theta=1\implies\cos\theta=√(1-\sin^2\theta)=√(1-\left(\frac12\right)^2)=\frac{\sqrt3}2`

Then

`\tan\theta=\tan\left(\sin^(-1)\frac12\right)=(\sin\theta)/(\cos\theta)=\frac{\frac12}{\frac{\sqrt3}2}=\frac1{\sqrt3}=\frac{\sqrt3}3`

as expected.

Answer 2

c. square root 3/3

Explanation:

just did it on edg