Graph polar coordinates and see the connection to the corresponding rectangular graph.[br]Note, you can get [math]\theta[/math] by clicking the [math]\alpha[/math] symbol in the input box or by holding "Alt + t"[br][br]You can move point [math]P[/math] either by dragging the slider, or by dragging the point P on the rectangular graph.[br][br]Key points to take away from this:[br]Each [b]vertical line[/b] on the rectangular graph corresponds to a [b]radial line[/b] on polar graph.[br]Each [b]horizontal line[/b] on the rectangular graph corresponds to a [b]circle[/b] on the polar graph.[br][br]The y-value on the rectangular graph tells us the radius on the polar graph when x (or [math]\theta[/math]) is a certain number.[br]e.g. The rectangular point [math]\left(\frac{\pi}{6},\frac{1}{2}\right)[/math] on the graph of [math]y=\sin x[/math] corresponds to the polar point [math]\left(\frac{1}{2},\frac{\pi}{6}\right)[/math] on the graph of [math]r=\sin\theta[/math].[br][br]
What does the graph of [math]r\cos\theta=3[/math] look like?[br]Solve for [math]r[/math] and graph above.
A vertical line through the polar point [math]\left(3,0\right)[/math][br][br]In general [math]r\cos\theta=d[/math] will give us a vertical line through [math]\left(d,0\right)[/math].
What does the graph of [math]r\sin\theta=2[/math] look like?[br]Solve for [math]r[/math] and graph above. (you may need to zoom out)
A horizontal line through the polar point [math]\left(2,\frac{\pi}{2}\right)[/math][br][br]In general [math]r\sin\theta=d[/math] will give us a vertical line through [math]\left(d,\frac{\pi}{2}\right)[/math].
What does [math]r=\cos\theta[/math] look like?
A circle centered at [math]\left(\frac{1}{2},0\right)[/math] with radius [math]\frac{1}{2}[/math].
What about [math]r=10\cos\theta[/math]?
A circle centered at [math]\left(5,0\right)[/math] with a radius of 5.[br][br]In general, [math]r=k\cos\theta[/math] will give us a circle centered at [math]\left(\frac{k}{2},0\right)[/math] with radius [math]\frac{k}{2}[/math]. Such a circle will also pass through the pole.