Question

How do you find sin (2theta) and cos (2theta) when given tan theta = -4 and sin theta <0 ?

Answer 1

`sin2theta=-8/17` and `cos2theta=-15/17`

Explanation:

We can use the identities `sin2theta=(2tantheta)/(1+tan^2theta)`

and `cos2theta=(1-tan^2theta)/(1+tan^2theta)`

As `tantheta=-4`

`sin2theta=(2*(-4))/(1+(-4)^2)=(-8)/17=-8/17`

and `cos2theta=(1-(-4)^2)/(1+(-4)^2)=(-15)/17=-15/17`

Observe that `sintheta<0` is superfluous information. As `tantheta<0`, `theta` lies in the interval `(pi/2,pi)` and hence `2theta` lies in interval `(pi,2pi)` and hence `sin2theta<0`. If `theta` lies in `((3pi)/2,2pi)`, `2theta` lies in interval `(3pi,4pi)` and again `sin2theta<0`.