Unit circle

This is an AoPSWiki Word of the Week for Oct 11-17

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 $(h,k)$ .

Trigonometry

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 above left are $\sin{\theta}$ and $\cos{\theta}$ , respectively. We use the pythagorean theorem to get:

$\sin^2{\theta}+\cos^2{\theta}=1$

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Unit circle

Trigonometry