Algebra 2 / Unit 8: Trigonometric Functions
8.1 Angles and Radian Measure
Notes
An angle in standard position starts on the positive $x$-axis. Counterclockwise is positive.
- A radian is the angle whose arc equals the radius. A full turn is $2\pi$ radians, so $180^\circ = \pi$ radians.
- Degrees to radians: multiply by $\displaystyle \frac{\pi}{180}$. Radians to degrees: multiply by $\displaystyle \frac{180}{\pi}$.
- Coterminal angles share a terminal side: add or subtract $360^\circ$ (or $2\pi$).
- Arc length: $s = r\theta$, with $\theta$ in radians.
- Example: $\displaystyle 60^\circ = \frac{\pi}{3}$, and on a circle of radius 9 it cuts an arc of $\displaystyle 9 \cdot \frac{\pi}{3} = 3\pi$.
Practice
Convert $150^\circ$ to radians.
$\displaystyle 150 \cdot \frac{\pi}{180} = \frac{5\pi}{6}$.
Convert $\displaystyle \frac{7\pi}{4}$ radians to degrees.
$\displaystyle \frac{7\pi}{4} \cdot \frac{180}{\pi} = 315$.
A circle has radius 6 cm. Find the arc length cut off by a central angle of $\displaystyle \frac{2\pi}{3}$.
$\displaystyle s = r\theta = 6 \cdot \frac{2\pi}{3} = 4\pi$. Using $120^\circ$ instead of radians gives 720.
Ready for more? Take the Unit 8 test.