Revision sheet

Parametric equations

• The point at time $t$ is $\big(x(t), y(t)\big)$.
• "Where at time $t$": evaluate. "When at $x = c$": solve $x(t) = c$.

Watch out: $x$ first, then $y$.

Eliminating the parameter

1. Solve one equation for $t$.
2. Substitute into the other.
3. With $\cos t$ and $\sin t$: use $\cos^2 t + \sin^2 t = 1$.

Watch out: solve for $t$; do not just replace $t$ by $x$.

Circles and ellipses

• $x = a\cos t$, $y = b\sin t$: half-axes $a$ (horizontal) and $b$ (vertical).
• Counterclockwise from $(a, 0)$; swapping $\sin$ and $\cos$ reverses it.
• $\cos(kt)$, $\sin(kt)$: one turn in $\dfrac{2\pi}{k}$.

Watch out: the number with $\cos$ in $x$ is the horizontal half-axis.

Polar coordinates

• $x = r\cos\theta$, $y = r\sin\theta$.
• $r = \sqrt{x^2 + y^2}$, $\tan\theta = \dfrac{y}{x}$.

Watch out: choose $\theta$ in the quadrant of the point.

Polar equations

• Convert with $x = r\cos\theta$, $y = r\sin\theta$, $r^2 = x^2 + y^2$.
• $r = a$: circle; $\theta = \alpha$: line; $r = 2a\sin\theta$: circle of radius $|a|$ through the origin.

Watch out: in $r = 2a\sin\theta$, $2a$ is the diameter.

Polar form of a complex number

• $z = r(\cos\theta + i\sin\theta)$.
• $r = \sqrt{a^2 + b^2}$; $\tan\theta = \dfrac{b}{a}$, quadrant of $(a, b)$.
• $a = r\cos\theta$, $b = r\sin\theta$.

Watch out: plot the point before choosing $\theta$.

Products and powers in polar form

• Product: multiply moduli, add arguments.
• De Moivre: $(r\,\mathrm{cis}\,\theta)^n = r^n\,\mathrm{cis}(n\theta)$.

Watch out: do not raise the real and imaginary parts separately.

Distance and midpoint in the complex plane

|z − w| and (z + w)/2.

Watch out: distance uses √(x² + y²), not x + y.