Geometric Representation of Euler's Formula

This applet enables the user to explore a geometric representation of Euler's formula, [math]e^{i\theta}=cos\theta+isin\theta[/math]. [br][br]The left window of this applet considers the left hand side of the formula, [math]e^{i\theta}[/math]. Recall that a defining property of the function [math]e^x[/math] is that it is its own derivative. Thus, [math]\frac{d}{dx}e^{kx}=ke^{kx}[/math] where [math]k[/math] is some constant. When we extend the exponential function to include imaginary numbers, this property must hold. Now, consider a particle moving in the complex plane whose position is defined parametrically by the function [math]z\left(t\right)=e^{it}[/math]. Then, its velocity, [math]v[/math], is given by [math]v\left(t\right)=\frac{dz}{dt}=ie^{it}[/math]. Note that the velocity is the position multiplied by [math]i[/math] and multiplying by [math]i[/math] is equivalent to rotating through a right angle. Since the initial position of the particle is [math]z\left(0\right)=1[/math] and its initial velocity is [math]v\left(0\right)=i[/math], the particle starts at [math]1+0i[/math] and begins moving upward. Using your mouse drag the blue point in the left window in the direction of [math]v[/math] in order to draw the motion of the particle. As you move the point, consider the following questions. [br][list][*]How would you describe the path the particle travels? [/*][*]Why does the particle travel along this path?[/*][/list]The right window of this applet considers the right hand side of the formula, [math]cos\theta+isin\theta[/math]. Using your mouse drag the black point in the right window around the unit circle in the complex plane. Toggle on/off the checkboxes in order to show/hide the real and imaginary parts as well as [math]cos\theta[/math] and [math]sin\theta[/math]. As you explore, consider the following questions. [br][list][*]What relationships exist between the real and imaginary parts and [math]cos\theta[/math] and [math]sin\theta[/math]?[/*][*]Why do these relationships exist? [/*][/list][br]After exploring both windows, consider how the windows relate to each other. How does this deepen your understanding of Euler's formula, [math]e^{i\theta}=cos\theta+isin\theta[/math]?

Information: Geometric Representation of Euler's Formula