Angles and Radian Measure

Foundational

Master standard position, radian definitions, and conversions between degrees and radians.

Angles and Radian Measure

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Convert $\frac{4\pi}{3}$ radians to degree measure.

$$\frac{4\pi}{3} \times \frac{180^\circ}{\pi} = 4 \times 60^\circ = 240^\circ$$

What are coterminal angles?

Cartesian coordinate plane illustrating coterminal angles, showing an angle theta in standard position alongside a positive full-rotation coterminal angle and a negative coterminal angle sharing the same terminal ray.

Angles in standard position that share the same terminal side. They differ by integer multiples of $360^\circ$ or $2\pi$ radians: $$\theta + 2\pi k, \quad k \in \mathbb{Z}$$

Find an angle coterminal with $\frac{11\pi}{4}$ in the interval $[0, 2\pi)$.

Subtract $2\pi$ (or $\frac{8\pi}{4}$): $$\frac{11\pi}{4} - \frac{8\pi}{4} = \frac{3\pi}{4}$$

What formula calculates the arc length $s$ intercepted by a central angle $\theta$ on a circle of radius $r$?

$$s = r\theta$$ Note: $\theta$ must be measured in radians.

Convert $150^\circ$ to exact radian measure.

$$150^\circ \times \frac{\pi}{180^\circ} = \frac{150\pi}{180} = \frac{5\pi}{6} \text{ radians}$$

What formula converts an angle from radians $R$ to degrees $D$?

$$D = R \times \frac{180^\circ}{\pi}$$

What is the geometric definition of one radian?

Diagram of a circle illustrating one radian as the central angle subtending a red arc whose length s equals the radius r.

One radian is the measure of a central angle that intercepts an arc whose length $s$ equals the radius $r$ of the circle: $$\theta = \frac{s}{r}$$

How many radians correspond to a complete rotation of $360^\circ$?

$$2\pi \text{ radians} = 360^\circ$$ Dividing both sides by $2$ gives the fundamental equivalence: $$\pi \text{ radians} = 180^\circ$$

What formula converts an angle from degrees $D$ to radians $R$?

$$R = D \times \frac{\pi}{180^\circ}$$

What is the radian measure of the four standard quadrantal angles in $[0, 2\pi)$?

A unit circle diagram showing the four standard quadrantal angles along the axes, labeled in radians and degrees: 0 rad (0 degrees), pi/2 rad (90 degrees), pi rad (180 degrees), and 3pi/2 rad (270 degrees).

$0^\circ = 0$ rad, $90^\circ = \frac{\pi}{2}$ rad, $180^\circ = \pi$ rad, and $270^\circ = \frac{3\pi}{2}$ rad.

What is the radian equivalent of the special acute angles $30^\circ$, $45^\circ$, and $60^\circ$?

$$30^\circ = \frac{\pi}{6}$$ $$45^\circ = \frac{\pi}{4}$$ $$60^\circ = \frac{\pi}{3}$$

Find an angle coterminal with $-120^\circ$ in the interval $[0^\circ, 360^\circ)$.

Add $360^\circ$: $$-120^\circ + 360^\circ = 240^\circ$$

If a circle has a radius of $6$ cm, find the arc length intercepted by a central angle of $\frac{2\pi}{3}$ radians.

$$s = r\theta = 6 \times \frac{2\pi}{3} = 4\pi \approx 12.57 \text{ cm}$$

What does it mean for an angle $\theta$ to be in standard position in the Cartesian plane?

Cartesian coordinate plane with an angle in standard position, showing its vertex at the origin, initial side on the positive x-axis, and a counterclockwise curved arrow to the terminal side in the first quadrant.

An angle is in standard position when its vertex is at the origin $(0, 0)$ and its initial side lies along the positive $x$-axis. Counterclockwise rotation yields a positive angle; clockwise rotation yields a negative angle.

Angles and Radian Measure

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