Notes
Radians
- $180^\circ = \pi$ radians. Multiply by $\displaystyle \frac{\pi}{180}$ to get radians and by $\displaystyle \frac{180}{\pi}$ to get degrees.
- Arc length $s = r\theta$ ($\theta$ in radians). Coterminal angles differ by $2\pi$.
Unit circle
An angle $\theta$ hits the unit circle at $(\cos\theta, \sin\theta)$.
| $\theta$ | $0$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ |
|---|---|---|---|---|---|
| $\cos\theta$ | $1$ | $\displaystyle \frac{\sqrt{3}}{2}$ | $\displaystyle \frac{\sqrt{2}}{2}$ | $\displaystyle \frac{1}{2}$ | $0$ |
| $\sin\theta$ | $0$ | $\displaystyle \frac{1}{2}$ | $\displaystyle \frac{\sqrt{2}}{2}$ | $\displaystyle \frac{\sqrt{3}}{2}$ | $1$ |
Use reference angles for other quadrants. Positive functions: I all, II sine, III tangent, IV cosine.
Graphs
For $y = a\sin(b(x - h)) + k$ or $y = a\cos(b(x - h)) + k$: amplitude $|a|$, period $\displaystyle \frac{2\pi}{b}$, midline $y = k$.
Identities
- $\sin^2\theta + \cos^2\theta = 1$, $\displaystyle \tan\theta = \frac{\sin\theta}{\cos\theta}$.
- $\displaystyle \csc\theta = \frac{1}{\sin\theta}$, $\displaystyle \sec\theta = \frac{1}{\cos\theta}$, $\displaystyle \cot\theta = \frac{1}{\tan\theta}$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle 240 \cdot \frac{\pi}{180} = \frac{4\pi}{3}$.
$\displaystyle \frac{5\pi}{3}$ is in Quadrant IV with reference angle $\displaystyle \frac{\pi}{3}$; cosine is positive there.
Period $\displaystyle = \frac{2\pi}{\pi} = 2$.
$\displaystyle \cos\theta = \frac{12}{13}$ (positive in Quadrant IV), so $\displaystyle \tan\theta = \frac{-5/13}{12/13} = -\frac{5}{12}$.
$\displaystyle \cos\theta \cdot \frac{\sin\theta}{\cos\theta} = \sin\theta$.
$\displaystyle -\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A Ferris wheel is 120 feet across and its center is 65 feet above the ground. It makes one full turn every 8 minutes. A rider boards at the lowest point at $t = 0$.
(a) Find the rider's maximum and minimum heights.
Show the answer and scoring guide
Radius $= 60$. Maximum $= 65 + 60 = 125$ feet; minimum $= 65 - 60 = 5$ feet.
Check off the points you earned:
(b) Write a function $h(t)$ for the rider's height after $t$ minutes.
Show the answer and scoring guide
Amplitude 60, midline 65, $\displaystyle b = \frac{2\pi}{8} = \frac{\pi}{4}$. Starting at the bottom: $\displaystyle h(t) = -60\cos\left(\frac{\pi}{4}t\right) + 65$.
Check off the points you earned:
(c) When does the rider first reach a height of 95 feet?
Show the answer and scoring guide
$\displaystyle -60\cos\left(\frac{\pi}{4}t\right) + 65 = 95 \Rightarrow \cos\left(\frac{\pi}{4}t\right) = -\frac{1}{2} \Rightarrow \frac{\pi}{4}t = \frac{2\pi}{3} \Rightarrow t = \frac{8}{3} \approx 2.67$ minutes.
Check off the points you earned: