How do you determine tantheta given costheta=1/8, (3pi)/2<theta<2pi?

2 Answers
Dec 19, 2017

-3sqrt7.

Explanation:

costheta=1/8 rArr sectheta=8.

:. tan^2theta=sec^2theta-1=8^2-1=63=9*7.

Hence, tantheta=sqrt63=+-3sqrt7.

But, given that, 3/2pi lt theta lt 2pi,

rArr tantheta=-3sqrt7.

Dec 19, 2017

tantheta=-3sqrt7

Explanation:

costheta=1/8

or,cos^2theta=1/64

or,1-sin^2theta=1/64color(blue)([[sin^2+cos^2theta=1,so,1-sin^2theta=cos^2theta]])

or,sin^2theta=1-1/64

or,sin^2theta=63/64

or,sintheta=+-sqrt63/8

Here sintheta=-sqrt63/8 ,As in fourth quarter the value of sintheta is negative

tantheta=sintheta/costheta=(-sqrt63/8/1/8=-sqrt63/cancel8xxcancel8)=-sqrt63

So,tantheta=-sqrt63=-3sqrt7