# Right Triangles on the Unit Circle

- Author:
- Ken Schwartz

- Topic:
- Circle, Right Triangles, Triangles, Unit Circle

Click and drag the point on the circle to change the right triangle that is formed. Check and uncheck the "Neg " box to see the effects of measuring in different directions (counterclockwise and clockwise). See below for more information.

The "

**" is a circle of radius 1, centered at the Origin. We establish the positive x-axis as the "initial side" of an angle, with any ray from the Origin forming its "terminal side". If we draw a***Unit Circle***segment**from the Origin to the point where the terminal side intersects the Circle, we can define that as the**Hypotenuse**of a right triangle. The triangle's**Opposite**side will extend from the intersection point vertically (up or down) to the x-axis. Its**Adjacent**side extends from the Origin to the intersection of the**Opposite**side and the x-axis.**We define the angle to extend counterclockwise from the**__Reference Angles:__**Initial Side**to the**Terminal Side**. However, if we are working with the right triangles described above, the angle in the triangle that we would normally call "" must be positive and cannot exceed 90 degrees. We define**Reference Angles**to be**the acute angle from the x-axis to the****HYP**. The**Reference Angle**is the "theta" of the right triangle.**Since we can continue around the circle as many times as we like, the angle can exceed 360 degrees ( radians). However, it is easiest to work with angles in the range of 0 - 360 (or 0 - radians). Therefore we define**__Coterminal Angles:__**Coterminal Angles**as angles that are at the same place on the circle, but could result from multiple rotations. For example, an angle of 10 degrees is the same as an angle of 370 degrees, since 370 degrees includes one full rotation of 360 degrees plus another 10 degrees. This angle could also be called 730 degrees (2 rotations + 10 deg), 3610 degrees (10 rotations + 10 deg), etc. If is the number of rotations, then (or ) is*coterminal with*. If we rotate in the clockwise direction, we consider to be negative. So, for example, an angle of 330 degrees (clockwise, 330 deg positive) is*coterminal with*-30 degrees (counterclockwise, 30 deg negative), since they end at the same point.**Co****terminal = "sharing the terminal side"**

**The**__Trig Functions on the Unit Circle:__**Hypotenuse**has a length of 1, since it is the radius of the Unit Circle. Since , , and**HYP=1**, we have the relationships: = y-coordinate of point at end of**HYP**= x-coordinate of point at end of**HYP**