Question

If sin theta is equal to 2/3, theta not in quadrant 1, find tan theta?

Answer 1

`tantheta=-2/sqrt5`

Explanation:

`"using the "color(blue)"trigonometric identities"`

`•color(white)(x)sin^2theta+cos^2theta=1`

`rArrcostheta=+-sqrt(1-sin^2theta)`

`•color(white)(x)tantheta=sintheta/costheta`

`"since "sintheta>0" but not in first quadrant then"`

`theta" is in second quadrant"`

`"where "costheta" and "tantheta<0`

`costheta=-sqrt(1-(2/3)^2)`

`color(white)(costheta)=-sqrt(1-4/9)=-sqrt(5/9)=-sqrt5/3`

`tantheta=(2/3)/(-sqrt5/3)=2/3xx-3/sqrt5=-2/sqrt5`

Answer 2

`tantheta=-2/sqrt5`

Explanation:

Here,

`sintheta=2/3 > 0=>I^(st)Quadrant or II^(nd)Quadrant`

`(i)`But given that , theta is not in quadrant 1

So, we have

`(ii)pi/2 < theta < pi toII^(nd)Quadrant`

`:.sintheta=2/3 >0 , costheta < 0and tantheta < 0`

Now , using trig.identity , we get

`cos^2theta=1-sin^2theta=1-4/9=5/9=(sqrt5/3)^2`

`=>costheta=-sqrt5/3 < 0`

Hence,

`tantheta=sintheta/costheta=(2/3)/(-sqrt5/3)=-2/sqrt5 < 0`