Answer:
tan(2x)=-24/7
Step-by-step explanation:
Since tan(x)=sin(x)/cos(x)
We are going to need sin(x) any time, so lets find it right away.
To do this, remember that. sin(x)^2 + cos(x)^2=1
so. [tex]sin(x)= \sqrt{1 - (cos(x)^{2} )}=\sqrt{1-(-4/5)^{2} }[/tex]
This leads to.
[tex]\sqrt{\frac{25-16}{25} } =\sqrt{\frac{9}{25} }=+/- 3-5[/tex]
We have obtained two solutions, -3/5 and 3/5.
We need to pick one, since not all of them are correct for our scenario, in this case, we've been told that x belongs to the range [180, 270], in this range, sin(x)<0.
So in our previous solution, we have that sin(x)= -3/5
Now, to find tang(2x), we need to apply the definition
Tang(2x)=sin(2x)/cos(2x).
Lets remember that
sin(2x)=2*sin(x)*cos(x)
cos(2x)=cos(x)^2 - sin(x)^2.
Lets evaluate our given result.
sin(2x)=2*(-3/5)*(4/5)=-24/25
cos(2x)=(-4/5)^2 - (3/5)^2=7/25
Hence
tan(2x)= -(24/25) / (7/25)=-24/7
