Unit Circle
Since the trigonometric ratios do not depend on the size of the triangle, you can always use a right-angled triangle where the hypotenuse has length one. In the drawing below is the graph of the unit circle on the `X - Y` coordinate Axis. It can be seen from the graph, that the Unit Circle is defined as having a radius equal to 1. The unit circle is fundamentally related to concepts in trigonometry. The trigonometric functions can be defined in terms of the unit circle, and in doing so, the domain of these functions is extended to all real numbers. Every point on the trigonometric circle corresponds to a right triangle with vertices at the origin and the point on the unit circle. The right triangle has leg lengths that are equal to the absolute values of the `x` and `y` coordinates, respectively. Since the hypotenuse of the right triangle is always 1, the values of the `x` and `y` coordinates of a point on the circle are always equal to, respectively, the cosine and sine of the angle `alpha`.
Created with GeoGebra by Vitor Nunes
Interactivity
Drag point A along the first quadrant and note the following:
The value of `cos alpha` is equal to the adjacent leg of the triangle formed from the intersection of the end side with the circumference (`x`-axis).
The value of `sin alpha` is equal to the opposite leg of the triangle formed from the intersection of the end side with the circumference (`y`-axis).
The value of `tan alpha` is equal to the opposite leg of the triangle formed from the intersection of the end side with the line x = 1 (gray).
Sine as Percentage
Nobody ever told me in my years of schooling: sine and cosine are percentages. They vary from -100% to +100%, or max negative to max positive. An absolute height isn’t helpful, but if your sine value is 0.97, I know you’re almost at the top of your dome. Pretty soon you’ll hit the max, then start coming down again. How do we compute the percentage? Simple: divide the current value by the maximum possible (the radius of the dome, aka the hypotenuse).
