Unit Circle Trigonometry
Drag the slider for to adjust the angle. Notice the coordinates of point on the unit circle.
Unit Circle Definition of Trig Functions
We can extend the definitions of sine, cosine and tangent (and their reciprocals) to angles of any measure, by using the unit circle.
On the unit circle, if the point has coordinates , then from the right triangle we can see that, in Quadrant I:
Therefore, the point where the terminal side of intersects the the unit circle will have coordinates , assuming the angle is in standard position.
Since the signs of the coordinates depend on the quadrant, the signs of sine, cosine, and tangent will also depend on the quadrant.
We can determine the coordinates of the point on the unit circle by using , the reference angle of . This is the acute angle formed between the terminal side of and the -axis. Completing the right triangle will get us the ratios for , which will be identical to the ratios for , except that we may need to attach a negative sign depending on the quadrant.
You can drag the slider for the angle and see how the red reference angle changes and where the right triangle is that will allow us to find the coordinates of the point on the unit circle and, in turn, the ratios of .