The unit circle is a simple and unifying idea. The unit circle is defined as a circle of radius one centered at the origin of the Cartesian Coordinate System. The length of one is chosen, because it is simple and makes some of the math easier later on; All circle have the same properties.
A circle can be used to describe an angle as a fraction of its circumference. By convention, angles are measured from the right-most edge of the circle, with the counter-clockwise direction as positive and the clockwise direction as negative. The standard unit of measure is called a radian and one full rotation around the circle is approximately 6.283 multiplied by the radius of the circle or radians for short. Click and drag the point on the interactive circle below to get an idea of how angles are measured using the radius of the circle.
Every point along the circumference of the circle describes a corresponding right-triangle. The x and y component of the point correlates to the adjacent and opposite sides of the triangle. There are two things to notice: First, every angle has a corresponding right triangle. And second, the sides of the triangle can be both positive or negative.
A brief recap of the notation before moving forward. Any point along the circumference of the circle can be described using the angle represented by the symbol θ (theta). The same point is also described by the coordinates (x,y) corresponding the the right triangle with a hypotenuse of one. The hypotenuse of length one is also the radius of the circle.
Now that we have established the basics of the unit circle, the next area to explore is the trigonometric functions sine and cosine. The functions are specific to right triangles and relate the angle of a right-triangle to the ratio of its sides.
In the case of the unit circle: the hypotenuse or radius will always be of length one, the adjacent side corresponds with the x coordinate, and the opposite side corresponds with the y coordinate. Substituting corresponding components into the functions defininitions simplifies both equations and creates a beautiful connection between the basic properties of the unit circle and the functions. Let’s first look at the Sine function:
Subsitituting the variable y for the opposite side of the right triangle and one for the hypotenuse shows that the sine function corresponds to the vertical component of the triangle. In the interactive above, the opposite side of the triangle and the output of the function is highlighted in red. A similar property can be illustrated for the cosine function:
Substituting in the variable x for the adjacent side of the right triangle and one for the hypotenuse shows that the cosine function corresponds to the horizontal component of the triangle. Again, the unit circle is show on the left and the graph of the function is shown on the right. The adjacent side of the right-triangle and the output of the function are both highlighted in red.
There are six total trigonometric functions, shown below, that relate to the geometry of the right-triangle: sine, cosine, tangent, cosecant, secant, and cotangent. The variable r is introduced for a more general form of all the equations. In the case of the unit circle, one can be subsituted for the radius (also hypotenuse) of the circle.
The interactive below demonstrates how the three standard functions relate to the unit circle:
Another well known property of the right triangle is Pythagorean’s Theorem which relates the squared-length of the adjacent, opposite, and hypotenuse sides together.
In the case of the unit circle: the adjacent side of the triangle, a, corresponds with x. The opposite side of the triangle, b, corresponds with y. And the hypotenuse, c, corresponds with the length 1. Substituting the corresponding values into the Pythagorean equation results in the equation below. Note, this is the equation of a circle with a radius of one in the Cartesian Coordinate system.
Next, recall the properties of the functions sine and cosine from above:
Substitute the functions sine and cosine in for x and y and the result is Pythagorean’s identity.
For now, this is where this article ends. Hopefully, at some later date, it will be extended. There are more ideas to connect and relate to the properties of the Unit Circle. Here are some links in the references below if you are interested.