Introduction to the Unit Circle in Trigonometry

What is the Unit Circle? Definition: A unit circle is a circle that has a radius of 1. Typically, especially in trigonometry, the unit circle is centered at the origin.

Angles in the Unit Circle When discussing an angle (pronounced “theta”) in the unit circle, we start from the positive x-axis and rotate counter-clockwise.

Problem of the Day If we let be 30°, what will our coordinates be on the unit circle? (x,y) = (?,?).

Let’s Draw a Triangle We can use our knowledge of special right triangles to help us solve this problem. Let’s use our 30° angle to draw a 30°- 60°- 90° triangle.

Find the Lengths of the Sides What is the length of the hypotenuse? The short leg? The long leg?

Find the Coordinate Now that we have these lengths, how does it help us find the coordinate? (x,y) = (, )

How Does This Relate to Trigonometry? The unit circle has some nice properties. Because the hypotenuse of any right triangle we draw will be 1, we get: – sin( ) = – cos( ) = – tan( ) =

Why Is This Helpful? This allows us to find the exact values of sin( ), cos( ), and tan( ) for certain values of. Based on our findings, we can say that: – sin(30°) = – cos(30°) = – tan(30°) =

Homework Using the same process that we just used, find the (x,y) coordinates on the unit circle for when = 45° and = 60°.