Question

If `sinu=-4/5` and `u` is in `Q3`, then find `sin2u,cos2u` and `tan2u`?

Answer 1

See below.

Explanation:

Using identities:

`color(red)(sin(2x)=2sinxcosx)`

`color(red)(cos(2x)=1-2sin^2x)`

`color(red)(tanx=sinx/cosx)`

`color(red)(cos(x)=sqrt(1/2(cos(2x)+1))`
..................................................................................................................................

`color(green)(cos(2u))=1-2sin^2(u)=1-2(-4/5)^2=color(blue)(-7/25)`

`color(green)(sin(2u))=2sin(u)cos(u)=2(-4/5)(sqrt(1/2((-7/25)+1)))`

`-> = 2(-4/5)(-sqrt(9/25))=color(blue)(24/25)` - as `u` is in `Q3`, `cosu` is negative.

`color(green)(tan(2u))=sin(2u)/cos(2u)=(24/25)/(-7/25)==color(blue)(-24/7)`