Question

How do you evaluate sin(arctan -4/3)?

Answer 1

Find arctan of -4/3 first.
Drawing the triangle on the unit circle is good for this.

Answer 2

I know how to solve for arctan positive 4/3 but I'm not sure how to solve it when it is negative.

Answer 3

tangent is opposite over adjacent, which means that to get -4/3, either the opposite side or the adjacent side must be negative.

Answer 4

I think because of domain and range issues, you need to take the fourth quadrant angle.

Answer 5

Right. Only the triangle in the fourth quadrant need be considered because the inverse tangent function is defined over the interval\[\frac{ - \pi }{ 2 }<x <\frac{ \pi }{ 2 }\]

Answer 6

By Pythagorean theorem the third side (hypotenuse side) is 5. Now what is the sine ratio of the angle the terminal arm forms with the x-axis? That ratio is your desired answer.