Right Triangles on the Unit Circle
- Ken Schwartz
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 "Unit Circle" 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 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. Reference Angles: We define the angle to extend counterclockwise from the 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. Coterminal Angles: 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 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.
Coterminal = "sharing the terminal side"Trig Functions on the Unit Circle: The 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