2x, given tanx=4 and cosx<0 what is cos2x and sin2x

If you place this angle on a rectangular grid

--  cos(x) < 0 means that the terminal side of theangle must be in either quadrant II or quadrant III

--  tan(x) > 0 means that the terminal side of theangle must be in either quadrant I or quadrant III.

Therefore, the terminal side of the angle is in quadrant III.

 

Since tan(x) = 4, the point in quadrant III can be (-1, -4).

This means that the radius is  sqrt( (-1)² + (-4)² ) = sqrt(17).

 

Therefore, sin(x)  =  -4/sqrt(17)  and  cos(x)  =  -1/sqrt(17).

 

You can use the formula:  sin(2x)  =  2·sin(x)·cos(x)

and any of the formulas for cos(2x); one of them being cos(2x)  =  1 - sin²(x).