Unit circle

Unit circle.png

A unit circle is a circle whose radius has length 1.

In the Cartesian coordinate system, an equation of the form $(x-h)^2+(y-k)^2=1$ defines a unit circle with center $(h,k)$.


Unit circle with triangle.png

An unit circle centered at the origin can be used to calculate values for the basic trigonometric functions. Suppose we draw a ray starting from the origin and meeting the positive x-axis with an angle of $\theta$. If we drop a perpendicular from the point of intersection between the ray and the circle, we have a right triangle with hypotenuse of $1$.

Using the definitions $\sin x = \frac{\text{opposite}}{\text{hypotenuse}}$ and $\cos x = \frac{\text{near}}{\text{hypotenuse}}$, we find that $\sin \theta = \frac{y}{1} = y$ and $\cos \theta = \frac{x}{1} = x$.

We can read off values for sine and cosine of an angle this way; we can draw the angle and approximate the x and y coordinates of the intersection.

We can also prove one of he fundamental theorems of trigonometry: $\sin^2{\theta}+\cos^2{\theta}=1$. The proof is as follows:

We see that the length and width of the triangle in the diagram at left are $\sin{\theta}$ and $\cos{\theta}$, respectively. We use the pythagorean theorem to get:


Complex numbers

On the complex plane, all solutions to the polynomial $x^n = 1$ lie upon the unit circle. These are referred to as the roots of unity.

In polar form, the solution to this polynomial can be expressed as $\cos \left(\frac{2\pi k}{n}\right) + i\sin \left(\frac{2\pi k}{n}\right)$, where $k = 0,1,2,\ldots n-1$. This is commonly written as $\mathrm{cis} \left(\frac{2\pi k}{n}\right)$. Additionally, the solution points form a regular $n$-gon on the unit circle.

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