Question

Find the exact value of tan((cos^-1)(12/13))

Answer 1

Let
`x=\cos^(-1)(12)/(13)`, so that
`\cos x=(12)/(13)`.

Recall that


`\cos^2x+\sin^2x=1\implies\sin x=\frac5{13}`

where we take the positive root because
`\cos\theta` is only invertible if
`0\le\theta<\pi`, and
`\cos\theta>0` only if
`0\le\theta<\frac\pi2`, which means
`\sin\theta>0`.

Now,


`\tan\left(\cos^(-1)(12)/(13)\right)=\tan x=(\sin x)/(\cos x)=\frac{\frac5{13}}{(12)/(13)}=\frac5{12}`