Question

I need help with this question please.

Answer 1

tan(2x)=-24/7

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.
`sin(x)= \sqrt{1 - (cos(x)^(2) )}=\sqrt{1-(-4/5)^(2) }`

This leads to.

`\sqrt{(25-16)/(25) } =\sqrt{(9)/(25) }=+/- 3-5`

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