Graph of e^(it)
Exploration
A demo of the graph of the complex-valued function is presented above. Note that there is a -axis, representing time, and two axes Re and Im, representing the real and imaginary axes of the Complex Plane.
Do the following to explore the symmetry between rotation and translation of the "coil" above.
1. Rotate the graph so that you plot the time and real part. What function's graph does this remind you of?
2. Rotate the graph so that you plot the time and imaginary part. What function's graph does this remind you of?
3. Interpret your answers to Step 1 and Step 2 in terms of Euler's Formula:
4. Grab the blue point and move it. This rotates the graph of the exponential by a particular angle.
Rotate the graph by 90 degrees counter-clockwise (or radians).
5. Grab the red point and move it. This shifts the graph along the -axis. Can you find a shift that makes the rotated graph and the shifted graph the same?
6. Recall that the complex exponential satisfies:
The Left-Hand-Side of the equation is the function that is a translation of by .
Multiplication by the complex number performs a counter-clockwise rotation of the Complex Plane by radians.
If we rotate by 90 degrees, the way that we can shift the graph to produce the same graph is so that is . (You can also add any multiple of , so also works.)
Did this agree with what you found earlier in Step 4?