If sin(x)=(1)/(2) what is cos(x) and tan(x)? Explain your steps in complete sentences.

Question

If sin(x)=
`(1)/(2)` what is cos(x) and tan(x)? Explain your steps in complete sentences.

Answer 1

The solution is not unique, but given that
`\sin x=\frac12`, by the Pythagorean identity we have two solutions for
`\cos x`:


`\sin^2x+\cos^2x=1\implies\cos x=\pm√(1-\sin^2x)=\pm\frac{\sqrt3}2`

Then from this, we have two corresponding solutions for
`\tan x`. By definition,


`\tan x=(\sin x)/(\cos x)`

and so we can have, in the case of
`\cos x>0`.


`\tan x=\frac{\frac12}{\frac{\sqrt3}2}=\frac1{\sqrt3}`

or
`\tan x=-\frac1{\sqrt3}` in the case that
`\cos x<0`.