The following graph shows a unit circle and a graph of the sine function.[br][br]The blue slider controls the magnitude of the angle [math]\alpha[/math], shown in radians and [br]located in standard position on the unit circle, with a range of [math]-2\pi[/math] to [math]2\pi[/math].[br][br]The blue diamond on the unit circle shows the angle [math]\alpha[/math] in standard position.[br][br]The green dot on the unit circle shows the angle [math](\pi+\alpha)[/math] in standard position.[br][br]What relationship exists between the blue diamond on the unit circle, and the blue diamond on [br]the sine graph?[br][br]Does this same relationship exist for the two green dots?

The sine of an angle is the [i]y[/i]-coordinate of the intercepted point on the unit circle, so the blue diamond [br]on the unit circle has coordinates [math](cos(\alpha),sin(\alpha))[/math]. [br][br]The blue diamond on the sine graph lies at the point [math](\alpha, sin(\alpha))[/math]. Its [i]y[/i]-coordinate [br]equals the [i]y[/i]-coordinate of the diamond on the unit circle. This same relationship exists for the two green dots.[br][br]Move the blue slider to various positions while observing the diamond and the dot on the unit circle. [br]- What symmetry do the diamond and the dot on the unit circle exhibit?[br]- What relationships do you see between their coordinates?[br][br]Now focus on the diamond and dot on the sine graph as you move the blue slider.[br]- Do they exhibit any symmetry?[br]- What relationships exist between their coordinates?[br][br]- What relationship exists between the coordinates of the two blue diamonds? Or the two green dots?[br][br]If you wish to use other applets similar to this, you may find an index of all my applets here: [url=https://mathmaine.wordpress.com/2010/04/27/geogebra/]https://mathmaine.com/2010/04/27/geogebra/[/url]