# Unit Circle x-axis Symmetry: Cosine

Author:
Whit Ford
The following graph shows a unit circle and a graph of the cosine function. The blue slider controls the magnitude of the angle , shown in radians and located in standard position on the unit circle, with a range of to . The blue diamond on the unit circle shows the angle in standard position. The green dot on the unit circle shows the angle in standard position. What relationship exists between the blue diamond on the unit circle, and the blue diamond on the cosine graph? Does this same relationship exist for the two green dots?
The cosine of an angle is the x-coordinate of the intercepted point on the unit circle, so the blue diamond on the unit circle has coordinates . The blue diamond on the cosine graph lies at the point . Its y-coordinate equals the x-coordinate of the diamond on the unit circle. This same relationship exists for the two green dots. Move the blue slider to various positions while observing the diamond and the dot on the unit circle. - What symmetry do the diamond and the dot on the unit circle exhibit? - What relationships do you see between their coordinates? Now focus on the diamond and dot on the cosine graph as you move the blue slider. - Do they exhibit any symmetry? - What relationships exist between their coordinates? - What relationship exists between the coordinates of the two blue diamonds? Or the two green dots? If you wish to use other applets similar to this, you may find an index of all my applets here: https://mathmaine.com/2010/04/27/geogebra/