The unit circle

A unit circle is an ordinary circle that is dimensionless; more precisely, the radius of this circle is equal to one. The unit circle is used to nicely represent the definitions of the various goniometric functions.

What is the unit circle

A unit circle is a circle that has a radius of length one, and the center of this circle is located at the center of the coordinate system, i.e. at the point [0, 0]. Look at the following figure:

The circle is further divided into four parts, which we call quadrants. The first quadrant is at the top right, the second at the top left, the third at the bottom left, and the fourth at the bottom right. Because we will use this circle to represent angles, the degrees are highlighted on the circle. Where there are zero degrees, that's usually where the arm of the angle starts and points upward. Therefore, the point [0, 1] is marked with a right angle.

Definition of sine with cosine on a circle

The unit circle is a very nice way to represent the various goniometric functions. First, we plot an angle on the unit circle, and then we show where we can read the values of each function on the unit circle.

We plotted the angle ASB (in red) on the unit circle. We have named this angle alpha. The arms of the angle intersect the unit circle at two points: A and B. The important point for us will be B. If we draw a line from B parallel to the axis of x (that's the dashed horizontal line in the figure), that line will intersect the axis of y at just one point. We will mark this point P_s. The length of the line SP_s (the green line) is equal to the product of the angle alpha.

Since we are moving in a unit circle that is centered at the origin of the coordinate system, the length of the line SP_s (the green line) is equal to the y-coordinate of the point P_s which is equal to the y-coordinate of the point B.

We can also read the cosine on the same circle. So we draw a line parallel to the axis y passing through the point B. This line will intersect the axis x at the point we mark P_c. The length of the line SP_c is then equal to the cosine of the angle alpha. Again, we can say that this value is equal to the x-coordinate of the point B and P_c.

Definition of tangent and cotangent

Just as we defined the sine and cosine on the unit circle, we can define the tangent and cotangent here. For clarity, we will first show where the tangent occurs on the unit circle. For this we will need another line. This will be the line that is parallel to the axis y and passes through the point A, or the point [1, 0]. In the following figure, this is the blue line:

This line intersects the semi-line (the arm of the angle) SB at one point, which we will label P_t. The distance of the line AP_t is then equal to the tangent of the angle alpha. Again, just take the y-coordinate of the point P_t and we also get the tangent of the angle alpha.

To represent the cotangent on the unit circle, we will need one more line. This time it will be a line that passes through the point [0, 1] and is parallel to the axis x. Again, it is highlighted in blue:

This line intersects the semi-line (the arm of the angle) SB at one point, which we will mark P_k. The distance of the line CP_k is then equal to the cotangent of the angle alpha. As always, we can just take the x-coordinate of the point P_k to get the cotangent of the angle.

The question is what happens if the angle alpha is greater than 90 degrees for the tangent and greater than 180 degrees for the cotangent, because in these cases the line does not cross the arm of the angle, see the following figure:

At this point, we make the semi-line SB a straight line, and this line already intersects the line p.

We would do the same in the tangent case.

When tangent and cotangent are not defined

We can notice one interesting thing about the unit circle. If the angle alpha is equal to 180 degrees, then the line p and the arm of the angle will never intersect because they will be parallel. How do we fix this problem? No way, the cotangent of 180 degrees is not defined. Similarly, the tangent of 90 degrees is not defined, because such an arm will be parallel to the axis x and so will be parallel to the line with which it should intersect. The following figure shows this, at least for the cotangent.