Unit Circle

In mathematics, a unit circle is a circle of unit radius, a circumference whose radius is 1. In other words it is the locus of points that have unit distance (modulus equal to 1) from the origin. In fact, in trigonometry the unit circle has center at the origin of the cartesian coordinate system in the Euclidean plane.

If (x, y) is a point on the unit circle of the first quadrant, then x and y are the lengths of the sides of a right triangle whose hypotenuse has length 1. So, for the Pythagorean theorem:

The trigonometric functions sine and cosine can be defined on the unit circle as follows:

The unit circle is the locus of points of the plane having a distance less than or equal to the unit from the center of the circle. So:

-1 ≤ y ≤ 1   and   -1 ≤ x ≤ 1

From here we are able to determine the domain of sine and cosine in the unit circle:

-1 ≤ sin (t) ≤ 1  and  -1 ≤ cos (t) ≤ 1

The unit circle provides an intuitive way to view the sine and cosine as periodic functions with the identities:

*( for every integer k)

These identities result from the fact that the coordinates x and y of a point on the unit circle remain the same increasing or decreasing the angle t for any number of turns (1 turn = 2π radians).

And here they are for every quadrant. With the correct sign* (plus or minus) as per Cartesian Coordinates. 

Note that cos is first and sin is second, so it goes (cos, sin):

And in radians:

*And this is the trend of the sign of sine and cosine in the unit circle:

But sine and cosine are not the only functions of the real variable t.

Below you can see the six trigonometric functions of  t :

Evaluating Trigonometric Functions Using the Unit Circle:            

It is important to recognize the radian measure of the standard angles related to:

located in quadrants II, III, and IV.

Here there is a useful video about the evaluation of trigonometric functions:

Other Facts Derived From the Unit Circle:

The y coordinates of points for θ and  −θ are the opposite, so:

sin(−θ) = −sin(θ )

Therefore sin(θ ) is an odd function, as well as functions csc, tan and cot. 

While functions cos and sec are even functions.

Marco Salvatore