Question

What co secant 0 if tan 0 =3/4

Answer 1

`\tan\theta=\frac34>0`


`\cot\theta=\frac1{\tan\theta}=\frac43>0`


`\csc^2\theta=1+\cot^2\theta`

`\csc\theta=\pm√(1+\left(\frac43\right)^2)=\pm\frac53`

Without any more information, it's impossible to tell whether the cosecant is positive or negative. For example, if
`\theta=\tan^(-1)\frac34`, then


`\tan\left(\tan^(-1)\frac34\right)=\frac34`

`\tan\left(\pi+\tan^(-1)\frac34\right)=\frac34`

but


`\csc\left(\tan^(-1)\frac34\right)=\frac53`

`\csc\left(\pi+\tan^(-1)\frac34\right)=-\frac53`