Question

Find sec(x) and csc(x) if tan(x) = 3 and cos > 0

Answer 1

Recall that

`1+\tan^2x=\sec^2x\implies\sec x=√(1+\tan^2x)`

We know to take the positive root here because
`\cos x>0`, which means
`\sec x>0` as well. Then

`\sec x=√(1+3^2)=\sqrt4=2\implies\cos x=\frac12`

We then have

`\tan x=(\sin x)/(\cos x)=(\sin x)/(\frac12)=3\implies\sin x=\frac32\implies\sec x=\frac23`