Table of Contents
Understanding Angle Measure
In trigonometry, an angle is formed by two rays (sides of the angle) sharing a common endpoint (the vertex). In this chapter we focus on how to measure angles, and in particular on two related ways of measuring them: degrees and radians.
You will later use these angle measures in trigonometric ratios, the unit circle, and trigonometric functions, so the goal here is to become comfortable switching between different viewpoints and units for angles.
Angles in Standard Position
For trigonometry, it is convenient to place angles in a standard way on the coordinate plane.
An angle is said to be in standard position when:
- Its vertex is at the origin $(0,0)$.
- Its initial side lies along the positive $x$-axis.
- Its terminal side is obtained by rotating the initial side about the origin.
The amount and direction of this rotation determine the angle.
- A positive angle is generated by rotating the initial side counterclockwise.
- A negative angle is generated by rotating the initial side clockwise.
For example:
- A $60^\circ$ angle in standard position is a counterclockwise rotation of $60^\circ$ from the positive $x$-axis.
- A $-45^\circ$ angle in standard position is a clockwise rotation of $45^\circ$ from the positive $x$-axis.
Angles larger than $360^\circ$ or less than $-360^\circ$ are also allowed; they simply represent rotations that go around the origin more than once.
Degrees as a Measure of Angle
The most familiar unit of angle measure is the degree.
A full revolution (one complete turn around a point) is defined to be:
$$360^\circ$$
From this:
- A straight angle (half a revolution) is $180^\circ$.
- A right angle (a quarter of a revolution) is $90^\circ$.
You should already be comfortable with these common degree measures; our main concern here is how they relate to radians and how angles can be measured in more than one unit.
Radian Measure: Definition and Idea
A radian is another way to measure angles. It is especially natural in trigonometry and calculus because it is directly tied to circles.
Consider a circle of radius $r$. Take an arc of the circle with length $s$. The angle $\theta$ (in radians) that subtends this arc at the center is defined by
$$\theta = \frac{s}{r} \quad \text{(radians)}.$$
Key points:
- The angle in radians is a ratio: arc length divided by radius.
- Because it is a ratio of two lengths, radian measure is dimensionless (it has no units like cm or m).
- If the arc length equals the radius ($s = r$), then
$$\theta = \frac{s}{r} = 1,$$
so that angle is 1 radian.
This definition ties together three related quantities:
- radius $r$,
- arc length $s$,
- central angle $\theta$ (in radians).
You will often use the relationship
$$s = r\theta \quad \text{(with $\theta$ in radians)}.$$
Radians and the Full Circle
The circumference of a circle of radius $r$ is
$$C = 2\pi r.$$
A full revolution around the circle corresponds to traversing an arc equal to the entire circumference: $s = 2\pi r$.
Using the definition of radian measure:
$$\theta_{\text{full}} = \frac{s}{r} = \frac{2\pi r}{r} = 2\pi.$$
So:
- A full revolution is $2\pi$ radians.
- A half revolution (a straight angle) is $\pi$ radians.
- A quarter revolution (a right angle) is $\dfrac{\pi}{2}$ radians.
This leads to the fundamental equivalence:
$$360^\circ = 2\pi \text{ radians}.$$
All degree–radian conversions come from this basic relationship.
Converting Between Degrees and Radians
Since
$$360^\circ = 2\pi \text{ rad},$$
dividing both sides by $2$ gives
$$180^\circ = \pi \text{ rad}.$$
This is the most convenient starting point for conversions.
From Degrees to Radians
To convert an angle $\alpha^\circ$ in degrees to radians, multiply by $\dfrac{\pi}{180^\circ}$:
$$\alpha^\circ = \alpha \cdot \frac{\pi}{180^\circ} \text{ rad}.$$
Examples:
- $90^\circ$:
$^\circ = 90 \cdot \frac{\pi}{180} = \frac{\pi}{2} \text{ rad}.$$ - $45^\circ$:
$^\circ = 45 \cdot \frac{\pi}{180} = \frac{\pi}{4} \text{ rad}.$$ - $60^\circ$:
$^\circ = 60 \cdot \frac{\pi}{180} = \frac{\pi}{3} \text{ rad}.$$ - $270^\circ$:
$0^\circ = 270 \cdot \frac{\pi}{180} = \frac{3\pi}{2} \text{ rad}.$$ - Negative angle, $-30^\circ$:
$$-30^\circ = -30 \cdot \frac{\pi}{180} = -\frac{\pi}{6} \text{ rad}.$$
From Radians to Degrees
To convert an angle $\theta$ (in radians) to degrees, multiply by $\dfrac{180^\circ}{\pi}$:
$$\theta \text{ rad} = \theta \cdot \frac{180^\circ}{\pi}.$$
Examples:
- $\dfrac{\pi}{3}$ radians:
$$\frac{\pi}{3} \cdot \frac{180^\circ}{\pi} = 60^\circ.$$ - $\dfrac{5\pi}{6}$ radians:
$$\frac{5\pi}{6} \cdot \frac{180^\circ}{\pi} = 5 \cdot 30^\circ = 150^\circ.$$ - $-\dfrac{\pi}{4}$ radians:
$$-\frac{\pi}{4} \cdot \frac{180^\circ}{\pi} = -45^\circ.$$ - $2\pi$ radians:
$\pi \cdot \frac{180^\circ}{\pi} = 360^\circ.$$
Common Angles in Both Units
In trigonometry, certain angles appear over and over again. Knowing their degree and radian measures by memory is extremely useful.
Some of the most common:
- $0^\circ = 0$
- $30^\circ = \dfrac{\pi}{6}$
- $45^\circ = \dfrac{\pi}{4}$
- $60^\circ = \dfrac{\pi}{3}$
- $90^\circ = \dfrac{\pi}{2}$
- $120^\circ = \dfrac{2\pi}{3}$
- $135^\circ = \dfrac{3\pi}{4}$
- $150^\circ = \dfrac{5\pi}{6}$
- $180^\circ = \pi$
- $210^\circ = \dfrac{7\pi}{6}$
- $225^\circ = \dfrac{5\pi}{4}$
- $240^\circ = \dfrac{4\pi}{3}$
- $270^\circ = \dfrac{3\pi}{2}$
- $300^\circ = \dfrac{5\pi}{3}$
- $315^\circ = \dfrac{7\pi}{4}$
- $330^\circ = \dfrac{11\pi}{6}$
- $360^\circ = 2\pi$
These angles will later correspond to “nice” points on the unit circle and to simple trigonometric values.
Coterminal Angles
Two angles are coterminal if they share the same initial side and terminal side when drawn in standard position.
You can obtain coterminal angles by adding or subtracting full revolutions.
In degrees:
- If $\alpha^\circ$ is an angle, then
$$\alpha^\circ + 360^\circ k$$
is coterminal with $\alpha^\circ$ for any integer $k$.
In radians:
- If $\theta$ is an angle in radians, then
$$\theta + 2\pi k$$
is coterminal with $\theta$ for any integer $k$.
Examples:
- $30^\circ$, $390^\circ$ ($30^\circ + 360^\circ$), and $-330^\circ$ ($30^\circ - 360^\circ$) are all coterminal.
- $\dfrac{\pi}{4}$ and $\dfrac{\pi}{4} - 2\pi = -\dfrac{7\pi}{4}$ are coterminal.
Recognizing coterminal angles helps when simplifying expressions, analyzing periodic behavior, and working with the unit circle.
Measuring Angles in Applications
The choice between degrees and radians depends on context:
- Degrees are common in everyday descriptions:
- navigation (bearings),
- carpentry and construction,
- many geometry problems.
- Radians are standard in higher mathematics and many scientific formulas:
- trigonometric functions in calculus (derivatives, integrals) must use radian measure for standard formulas to hold,
- arc length and sector area formulas usually assume radians,
- many physical formulas involving oscillations, waves, and rotations naturally use radians.
The same angle can always be expressed in either unit; what changes is which unit makes the formulas simplest.
Summary of Key Relationships
- Definition (radian measure):
$$\theta = \frac{s}{r} \quad (\text{radians}), \quad s = r\theta.$$ - Full revolution:
$0^\circ = 2\pi \text{ rad}, \quad 180^\circ = \pi \text{ rad}.$$ - Conversion formulas:
- Degrees to radians:
$$\alpha^\circ = \alpha \cdot \frac{\pi}{180^\circ} \text{ rad}.$$ - Radians to degrees:
$$\theta \text{ rad} = \theta \cdot \frac{180^\circ}{\pi}.$$ - Coterminal angles:
- Degrees: $\alpha^\circ + 360^\circ k$.
- Radians: $\theta + 2\pi k$.
These ideas about angles and radians form the foundation for all later work in trigonometry, especially when studying trigonometric ratios, the unit circle, and trigonometric functions.