Table of Contents
Understanding Radians
In this chapter, we focus on measuring angles in radians. The general idea of angles and degrees is handled in the parent sections; here we zoom in on what radians are, how they relate to degrees, and how to work with them.
What a Radian Measures
A radian is a way to measure an angle using the arc length it cuts off on a circle.
Consider a circle of radius $r$ and an angle $\theta$ at the center. Let the length of the arc on the circle corresponding to this angle be $s$.
By definition, the angle $\theta$ in radians is
$$
\theta = \frac{s}{r}.
$$
So:
- $s$ is the length of the arc,
- $r$ is the radius of the circle,
- $\theta$ is the angle in radians.
An angle of 1 radian is the angle that cuts off an arc whose length equals the radius:
$$
\theta = 1 \text{ rad} \quad \Longleftrightarrow \quad s = r.
$$
A key point: the formula $\theta = s/r$ works for any circle radius $r$. That means that a radian is an absolute measure of angle, not tied to a specific circle size.
Radians and the Circle
The circumference of a circle with radius $r$ is
$$
C = 2\pi r.
$$
If we consider a full turn around the circle (one complete revolution), the corresponding arc length is the entire circumference, so for a full turn the angle in radians is
$$
\theta_{\text{full turn}} = \frac{C}{r} = \frac{2\pi r}{r} = 2\pi.
$$
So:
- A full revolution $= 2\pi$ radians.
- A half-turn (straight angle) $= \pi$ radians.
- A quarter-turn (right angle) $= \dfrac{\pi}{2}$ radians.
These values come up constantly in trigonometry.
Relationship Between Degrees and Radians
A full turn is also $360^\circ$. Since the same angle corresponds to both $360^\circ$ and $2\pi$ radians, we have:
$$
2\pi \text{ radians} = 360^\circ.
$$
Dividing both sides by $2$:
$$
\pi \text{ radians} = 180^\circ.
$$
This is the basic bridge between degrees and radians.
Converting Degrees to Radians
To convert an angle $\alpha$ in degrees to radians, use
$$
\alpha^\circ = \alpha \cdot \frac{\pi}{180} \text{ radians}.
$$
So, multiply the degree measure by $\dfrac{\pi}{180}$.
Examples:
- $180^\circ = 180 \cdot \dfrac{\pi}{180} = \pi$ radians.
- $90^\circ = 90 \cdot \dfrac{\pi}{180} = \dfrac{\pi}{2}$ radians.
- $45^\circ = 45 \cdot \dfrac{\pi}{180} = \dfrac{\pi}{4}$ radians.
- $30^\circ = 30 \cdot \dfrac{\pi}{180} = \dfrac{\pi}{6}$ radians.
- $60^\circ = 60 \cdot \dfrac{\pi}{180} = \dfrac{\pi}{3}$ radians.
Converting Radians to Degrees
To convert an angle $\theta$ in radians to degrees, use
$$
\theta \text{ radians} = \theta \cdot \frac{180^\circ}{\pi}.
$$
So, multiply the radian measure by $\dfrac{180^\circ}{\pi}$.
Examples:
- $\pi$ radians $= \pi \cdot \dfrac{180^\circ}{\pi} = 180^\circ$.
- $\dfrac{\pi}{2}$ radians $= \dfrac{\pi}{2} \cdot \dfrac{180^\circ}{\pi} = 90^\circ$.
- $\dfrac{\pi}{3}$ radians $= 60^\circ$.
- $\dfrac{\pi}{4}$ radians $= 45^\circ$.
- $\dfrac{\pi}{6}$ radians $= 30^\circ$.
- $2\pi$ radians $= 360^\circ$.
Common Angles in Radians
Because trigonometry frequently uses certain standard angles, it is useful to know their radian measures by heart.
For one full turn:
- $0^\circ = 0$,
- $30^\circ = \dfrac{\pi}{6}$,
- $45^\circ = \dfrac{\pi}{4}$,
- $60^\circ = \dfrac{\pi}{3}$,
- $90^\circ = \dfrac{\pi}{2}$,
- $120^\circ = \dfrac{2\pi}{3}$,
- $135^\circ = \dfrac{3\pi}{4}$,
- $150^\circ = \dfrac{5\pi}{6}$,
- $180^\circ = \pi$,
- $210^\circ = \dfrac{7\pi}{6}$,
- $225^\circ = \dfrac{5\pi}{4}$,
- $240^\circ = \dfrac{4\pi}{3}$,
- $270^\circ = \dfrac{3\pi}{2}$,
- $300^\circ = \dfrac{5\pi}{3}$,
- $315^\circ = \dfrac{7\pi}{4}$,
- $330^\circ = \dfrac{11\pi}{6}$,
- $360^\circ = 2\pi$.
These values will reappear often, especially in the unit circle and trigonometric functions.
Radians as a “Natural” Angle Measure
Radians are especially important in more advanced parts of trigonometry and calculus because they interact cleanly with arc length and with trigonometric functions.
Using the definition
$$
\theta = \frac{s}{r},
$$
we can always say:
- $s = r\theta$ when $\theta$ is measured in radians.
This simple formula holds only if $\theta$ is in radians. If $\theta$ were in degrees, an extra conversion factor would be needed.
Similarly, in calculus, formulas like
$$
\frac{d}{dx}(\sin x) = \cos x
$$
are true in this clean form only when $x$ is measured in radians. This is one of the key reasons radians are the standard in higher mathematics.
Working with Large and Small Radian Measures
Angles do not need to be between $0$ and $2\pi$. For example:
- $5\pi$ radians is more than one full turn:
- One full turn is $2\pi$, so $5\pi$ is $2\pi + 2\pi + \pi$ (two full turns plus another half-turn).
- $-\dfrac{\pi}{3}$ radians is a clockwise turn of $60^\circ$.
To interpret any radian angle:
- You can add or subtract multiples of $2\pi$ to find a coterminal angle between $0$ and $2\pi$ (or between $-\pi$ and $\pi$, depending on preference).
- Convert it to degrees if that helps your intuition.
Example:
- $7\pi/4$ radians:
- As degrees: $7\pi/4 \cdot \dfrac{180^\circ}{\pi} = 7 \cdot 45^\circ = 315^\circ$.
- $-3\pi/2$ radians:
- Add $2\pi$: $-3\pi/2 + 2\pi = \pi/2$, which is $90^\circ$,
- So $-3\pi/2$ is a $90^\circ$ turn clockwise.
Measuring Angles on the Unit Circle
The unit circle has radius $1$, so $r = 1$. On this special circle, the radian measure of an angle equals the length of the arc it cuts off:
$$
\theta = \frac{s}{1} = s.
$$
So on the unit circle:
- The arc length from angle $0$ to angle $\theta$ is exactly $\theta$ (if $\theta$ is in radians).
- For example, from $0$ to $\pi/3$ radians, the arc length is $\pi/3$.
This tight connection between angles and arc length on the unit circle is a major reason radians are standard in trigonometry.
Summary of Key Facts About Radians
- Definition: $\theta$ radians $= \dfrac{\text{arc length}}{\text{radius}}$.
- Full turn: $2\pi$ radians.
- Bridge: $\pi$ radians $= 180^\circ$.
- Degree to radian: multiply by $\dfrac{\pi}{180}$.
- Radian to degree: multiply by $\dfrac{180^\circ}{\pi}$.
- On a circle: $s = r\theta$ (with $\theta$ in radians).
- On the unit circle: angle (in radians) equals arc length.
These ideas about radians form the foundation for the unit circle, trigonometric functions, and many formulas you will encounter in trigonometry and beyond.