Question

If `sectheta+tantheta=3/2`, what is the value of `sintheta`?

Answer 1

`sin theta=5/13`

Explanation:

`tan theta +sec theta =3/2`
No idea-well do something!!
`sin theta/cos theta+1/cos theta=3/2`
And then the next stage becomes clear
`(sin theta+1)/cos theta=3/2`
`2(sin theta +1)=3cos theta`

So what now.? Well we only want to know about `sin theta`
And we do know `sin^2 theta +cos^2 theta=1`
So square both sides

`4(sin theta+1)^2=9cos^2 theta`
`4(sin^2 theta +2sin theta +1)=9(1-sin^2 theta)`
`4sin^2 theta +8sin theta+4=9-9 sin^2 theta`
`13sin^2theta+8sin theta-5=0`
Factorise
`(13 sin theta-5)(sintheta +1)=0`
`sintheta=5/13` or `sintheta=-1`
Only the first will do because if `sintheta =-1` then `costheta=0` and clearly we cannot divide by zero.

Answer 2

Given

`sectheta+tantheta=1.5=15/10=3/2`

`=>sectheta+tantheta=3/2……………(1)`

Again we know

`sec^2theta-tan^2theta=1……………(2)`

Dividing (2) by (1) we get

`sectheta-tantheta=2/3……………(3)`

Adding (1) and (3) we get

`2sectheta=3/2+2/3=13/6`

`=>sectheta=13/12`

Subtracting (3) from (1) we get

`2tantheta=3/2-2/3=5/6`

`=>tantheta=5/12`

`=>sinthetaxxsectheta=5/12`

`=>sinthetaxx13/12=5/12`

`=>sintheta=5/12xx12/13=5/13`

Alternative

`sectheta+tantheta=3/2`

`=>1/costheta+sintheta/costheta=3/2`

`=>(1+sintheta)/costheta=3/2`

`=>(1+sintheta)/sqrt(1-sin^2theta)=3/2`

`=>(sqrt(1+sintheta)sqrt(1+sintheta))/(sqrt(1-sintheta)sqrt(1+sintheta))=3/2`

for `sintheta!=-1`

`=>(sqrt(1+sintheta))/(sqrt(1-sintheta))=3/2`

`=>(1+sintheta)/(1-sintheta)=9/4`

`=>4+4sintheta=9-9sintheta`

`=>13sintheta=5`

`=>sintheta=5/13`