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

2 Answers

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.

Dec 12, 2016

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