AP Precalculus · Summary

Unit 3 summary

1. Radians measure arc over radius, $\theta = \frac{s}{r}$, and $\pi$ radians is 180 degrees. A periodic function repeats, so subtract whole periods to find a far-off value.
2. On the unit circle the point at angle $\theta$ is $(\cos\theta, \sin\theta)$, and $\tan\theta$ is the slope of the ray. Use the reference angle for the size and the quadrant for the sign.
3. For $a\sin(b(x - c)) + d$: amplitude $|a|$, midline $y = d$, period $\frac{2\pi}{|b|}$ and phase shift c. Factor b out before you read c.
4. Amplitude is half of the maximum minus the minimum, the midline is their average, and a maximum and the next minimum are half a period apart.
5. Arcsin and arctan return angles between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, and arccos returns angles from 0 to $\pi$. Look for a second solution in each period.
6. Use $\sin^2\theta + \cos^2\theta = 1$ and the sum and double-angle identities to rewrite expressions; $\sin(\alpha + \beta)$ is not $\sin\alpha + \sin\beta$ in general.
7. Polar to rectangular: $x = r\cos\theta$ and $y = r\sin\theta$. The distance from the origin is |r|, and a negative r points the opposite way.