Definition of the unit circle

A unit circle is a circle whose radius is equal to 1. Furthermore, the circle has its center at the origin of a rectangular coordinate system.

Unit circle

Let P = (x , y) be a point on the circle. Then, make a right triangle by drawing a line perpendicular to x. The line is shown in green.

Unit circle

The horizontal leg of the triangle is x units away from the origin and the vertical leg of the triangle is y units away from the origin.

Unit circle

Using the pythagorean theorem, we get x2 + y2 = 12    or    x2 + y2 = 1

We can take a step further so we can derive an important trigonometric function. Let us name t the angle made with the radius in red and the x axis.

Unit circle

Take a close look at the triangle and you will see as we learned before that the adjacent side to angle t is x and the opposite side is y.

Therefore, sin(t) = y / 1 = y   and cos(t) = x / 1 = x

sin(t) is read as sine of t and sin(t) is the y-coordinate of point P.

cos(t) is read as cosine of t and cos(t) is the x-coordinate of point P.

We can replace x = cos(t) and sin(t) in x2 + y2 = 1

We get cos(t)2 + sin(t)2 = 1

Definitions of the six trigonometric functions in terms of the unit circle.

cos(t) = x   

sec(t) = 1/x, with x not equal to 0 ( sec(t) is the reciprocal of cos(t) )

sin(t) = y       

csc(t) = 1/y, with y not equal to 0 ( csc(t) is the reciprocal of sin(t) )

tan(t) = y/x, with x not equal to 0

cot(t) = x/y, with y not equal to 0 ( cot(t) is the reciprocal of tan(t) )

We can also call the six trigonometric functions above circular functions because of the fact that we are using a unit circle to express the functions values.

We just derived some of the most important trigonometric identities. We have just scratched the surface of what we can do with the unit circle. Next lesson will show that the unit circle can also be used to find sin (45 degrees)

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