Notes
Measuring angles
- An angle in standard position has its vertex at the origin and its initial side on the positive $x$-axis. Counterclockwise is positive.
- Convert with $180^\circ = \pi$ radians: multiply by $\displaystyle \frac{\pi}{180}$ for radians, or by $\displaystyle \frac{180}{\pi}$ for degrees.
- Coterminal angles differ by multiples of $360^\circ$ or $2\pi$. The reference angle is the acute angle to the $x$-axis.
Circles
- Arc length $s = r\theta$ and sector area $\displaystyle A = \frac{1}{2}r^2\theta$, with $\theta$ in radians.
Rotational motion
- Angular speed $\displaystyle \omega = \frac{\theta}{t}$ and linear speed $v = r\omega$.
- One revolution equals $2\pi$ radians.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle \frac{240}{180} = \frac{4}{3}$.
$\displaystyle -\frac{3\pi}{4} \cdot \frac{180^\circ}{\pi} = -135^\circ$.
Add $\displaystyle 2\pi = \frac{12\pi}{6}$: $\displaystyle \frac{5\pi}{6} + \frac{12\pi}{6} = \frac{17\pi}{6}$.
$-120^\circ$ is coterminal with $240^\circ$ in Quadrant III, and $240^\circ - 180^\circ = 60^\circ$.
$s = r\theta = 8 \cdot 2.5 = 20$.
$\displaystyle \omega = \frac{2\pi}{8} = \frac{\pi}{4}$ rad/s, so $\displaystyle v = 4 \cdot \frac{\pi}{4} = \pi$ m/s.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A Ferris wheel has radius 20 m and makes one full revolution every 40 seconds.
(a) Find the angular speed of the wheel in radians per second.
Show the answer and scoring guide
$\displaystyle \omega = \frac{2\pi}{40} = \frac{\pi}{20}$ rad/s.
Check off the points you earned:
(b) Find the linear speed of a rider at the rim.
Show the answer and scoring guide
$\displaystyle v = r\omega = 20 \cdot \frac{\pi}{20} = \pi$ m/s, about $3.14$ m/s.
Check off the points you earned:
(c) How far does a rider travel along the rim in 10 seconds?
Show the answer and scoring guide
In 10 s the angle is $\displaystyle \theta = \frac{\pi}{20} \cdot 10 = \frac{\pi}{2}$, so $\displaystyle s = r\theta = 20 \cdot \frac{\pi}{2} = 10\pi$ m, about $31.4$ m.
Check off the points you earned: