The reference angle is simply the acute angle between the terminal side and the x-axis
The terminal angles in the four quadrants are
- [tex]\bar \theta = \theta[/tex].
- [tex]\bar \theta = \pi - \theta[/tex].
- [tex]\bar \theta = \theta - \pi[/tex].
- [tex]\bar \theta = 2\pi - \theta[/tex]
The given parameters are:
[tex]\theta \to[/tex] reference angle
[tex]\bar \theta \to[/tex] terminal angle
(a) If the reference angle is in the first quadrant
In the first quadrant, the x-axis is at [tex]0\ rad[/tex]
So, the terminal angle is:
[tex]\bar \theta = \theta - 0[/tex]
[tex]\bar \theta = \theta[/tex]
(b) If the reference angle is in the second quadrant
In the second quadrant, the x-axis is at [tex]\pi \ rad[/tex],
While the reference angle is between [tex]\frac{\pi}{2}[/tex] and [tex]\pi \ rad[/tex]
It means that, we have to subtract the reference angle from the x-axis
So, the terminal angle is:
[tex]\bar \theta = \pi - \theta[/tex]
(c) If the reference angle is in the third quadrant
Here, the x-axis is still at [tex]\pi \ rad[/tex],
But the reference angle is between [tex]\pi \ rad[/tex] and [tex]\frac{3}{2}\pi[/tex]
It means that, we have to subtract the x-axis from the reference angle.
So, the terminal angle is:
[tex]\bar \theta = \theta - \pi[/tex]
(d) If the reference angle is in the fourth quadrant
Here, the x-axis is at [tex]2\pi \ rad[/tex]
And the reference angle is between [tex]\frac{3}{2}\pi[/tex] and [tex]2\pi \ rad[/tex]
It means that, we have to subtract the reference angle from the x-axis.
So, the terminal angle is:
[tex]\bar \theta = 2\pi - \theta[/tex]
See attachment for illustration of terminal angles
Read more about terminal angles at:
https://brainly.com/question/12891381
