Notes
Angles and the unit circle
- $180^\circ = \pi$ radians. Convert with $\displaystyle \frac{\pi}{180}$ or $\displaystyle \frac{180}{\pi}$.
- On the unit circle the point at angle $\theta$ is $(\cos\theta, \sin\theta)$. Use reference angles and quadrant signs.
- Arc length $s = r\theta$ and sector area $\displaystyle A = \frac{1}{2}r^2\theta$ (radians only).
Graphs of sine and cosine
- For $y = A\sin(B(x - C)) + D$: amplitude $|A|$, period $\displaystyle \frac{2\pi}{|B|}$, phase shift $C$ (right), midline $y = D$.
- Maximum $= D + |A|$, minimum $= D - |A|$.
Modeling
- From a maximum $M$ and minimum $m$: amplitude $\displaystyle \frac{M - m}{2}$, midline $\displaystyle \frac{M + m}{2}$. From a period $P$: $\displaystyle B = \frac{2\pi}{P}$.
- Choose cosine if the data starts at a maximum, sine if it starts at the midline.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle 300 \cdot \frac{\pi}{180} = \frac{5\pi}{3}$.
$\displaystyle \frac{11\pi}{6}$ is in quadrant IV with reference angle $\displaystyle \frac{\pi}{6}$, where sine is negative.
Amplitude is $|{-5}| = 5$. Period $\displaystyle = \frac{2\pi}{\pi/3} = 6$.
The midline is $-1$ and the amplitude is 2, so the values go from $-3$ to $1$.
$\displaystyle A = \frac{1}{2}(100)\left(\frac{\pi}{5}\right) = 10\pi$.
Factor out 3: $\displaystyle \sin(3(x - \frac{\pi}{3}))$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
The depth of water in a harbor varies with the tides. High tide is 9 m and low tide is 3 m. The time between one high tide and the next is 12 hours. At $t = 0$ it is high tide.
(a) Find the amplitude and the midline of the depth function $D(t)$.
Show the answer and scoring guide
Amplitude $\displaystyle = \frac{9 - 3}{2} = 3$. Midline $\displaystyle = \frac{9 + 3}{2} = 6$.
Check off the points you earned:
(b) Write a cosine function for $D(t)$.
Show the answer and scoring guide
$\displaystyle B = \frac{2\pi}{12} = \frac{\pi}{6}$. Since $t = 0$ is a maximum, $\displaystyle D(t) = 3\cos\left(\frac{\pi}{6}t\right) + 6$.
Check off the points you earned:
(c) Find the depth at $t = 2$ hours.
Show the answer and scoring guide
$\displaystyle D(2) = 3\cos\left(\frac{\pi}{3}\right) + 6 = 3(0.5) + 6 = 7.5$ m.
Check off the points you earned: