Question

Question #a9735

Answer 1

`sin 2theta=24/25, cos 2theta=-7/25, tan 2theta=-24/7`

Explanation:

It is known that
`cos theta =3/5`
And
`0< theta<90^@`

We also know that
`sin theta = sqrt (1-cos^2 theta)`
Since `theta` is an angle of the first quadrant `sin theta>0` and we have
`sin theta =(+)sqrt(1-(3/5)^2)=sqrt(1-9/25)=sqrt(16/25)=4/5`

It's also known that

`sin 2theta=2sin theta *cos theta`
=> `sin 2theta=2*4/5*3/5=24/25`
And that
`cos 2theta=cos^2 theta-sin^2 theta=(3/5)^2-(4/5)^2=(9-16)/25=-7/25`

So

`tan 2theta=(sin 2theta)/(cos theta)=(24/cancel(25))/(-7/cancel(25))=-24/7`