[size=150]Suppose there is a point [math]P[/math] on the unit circle at [math](1,0)[/math].[/size][br][img]data:image/png;base64,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[/img][br]Describe how the [math]x[/math]-coordinate of [math]P[/math] changes as it rotates once counterclockwise around the circle.[br]
Describe how the [math]y[/math]-coordinate of [math]P[/math] changes as it rotates once counterclockwise around the circle.[br]
[size=150]Use the class display, the table from a previous lesson, or the applet to estimate the value of [math]y=\cos\left(\theta\right)[/math] and [math]y=\sin\left(\theta\right)[/math] where [math]\theta[/math] is the measure of an angle in radians.[/size]
What do you notice about the two graphs?
Explain why any angle measure between 0 and [math]2\pi[/math] gives a point on each graph.
Could these graphs represent functions? Explain your reasoning.
[size=150]Looking at the graphs of [math]y=\cos\left(\theta\right)[/math] and [math]y=\sin\left(\theta\right)[/math], at what values of [math]\theta[/math] do [math]\cos\left(\theta\right)=\sin\left(\theta\right)[/math]? [br][/size]
Where on the unit circle do these points correspond to?
[size=150]For each of these equations, first predict what the graph looks like, and then check your prediction using the applet at the end of the activity. [br][/size][br][math]y=\cos\left(\theta\right)+\sin\left(\theta\right)[/math]
[math]y=\cos^2\left(\theta\right)[/math]
[math]y=\sin^2\left(\theta\right)[/math]
[math]y=\cos^2\left(\theta\right)+\sin^2\left(\theta\right)[/math]
To enter the symbol for [math]\theta[/math], use the onscreen keyboard which can be accessed through the keyboard [img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACoAAAAdCAYAAADcvP5OAAABNUlEQVRYCe1XSw6FIAz01C71Gi49hx5Bd57BK7jlZUxqBl9KCnloeNqEMDS1HVo+UrlCpCqEp/sium2bm+fZjeOYtSEGYlnFI7quq2vb1tV1fUlDLMS0iEe07/tLCHIiENMiHlF2cCV+DtFhGI7JAiPLLBin2nDF2KeGg6VPJcHBtMk8kyjPOifmCmg4WPqu6/Y1iCUALGXEGA3jVBueuEaO9UGi7xo93XycOQ0HM4qyyp0vpZcxeim96GJsflp6dpYTa1lkfTCjOcmxbyak4WiisuOtPRPSsEaO9dFE+WML1six3uLnf4nKDrf2nDkNR2dUjhfNYQ590o8zngVN0+znYw5SZ5+IlfQUQQnw4Jqm6bjjrbs71g4xkh93lrVyl021LIsroXnH013ZssR9iVqyFGPzAXPwxRp5+prYAAAAAElFTkSuQmCC[/img] icon. You should enter each equation as a function. For example, [math]y=\sin(theta)[/math] can be entered as [math]f(\theta)=\sin(\theta)[/math]. Some examples are given in the applet below.
[size=150]For the equation given, predict what the graph looks like, and then check your prediction using technology: [math]y=\theta+\cos\left(\theta\right)[/math].[/size]