Table of Contents
Measuring angles in degrees
In trigonometry, one of the main ways to measure the size of an angle is with degrees. This chapter focuses only on the degree system; the separate chapter on radians will cover that system and the relationship between them.
An angle measured in degrees uses the symbol $^\circ$ (read “degrees”). For example:
- $30^\circ$ is “thirty degrees”
- $90^\circ$ is “ninety degrees”
A full turn around a point is defined to be:
$$360^\circ$$
From this basic fact, other common angles are defined.
Key reference angles in degrees
Some angles appear so often in trigonometry that it is useful to remember them:
- Right angle: $90^\circ$
This is the angle formed by perpendicular lines. - Straight angle: $180^\circ$
This forms a straight line. - Full (or complete) angle: $360^\circ$
One full rotation.
Common smaller angles:
- $30^\circ$
- $45^\circ$
- $60^\circ$
These are especially important later when you work with exact trigonometric values and the unit circle.
Types of angles (by degree measure)
Here we classify angles specifically by their degree measure.
- Acute angle: an angle that is greater than $0^\circ$ and less than $90^\circ$
$ ^\circ < \theta < 90^\circ$$
- Right angle: an angle exactly equal to $90^\circ$
$$\theta = 90^\circ$$ - Obtuse angle: an angle that is greater than $90^\circ$ and less than $180^\circ$
$^\circ < \theta < 180^\circ$$ - Straight angle: an angle exactly equal to $180^\circ$
$$\theta = 180^\circ$$ - Reflex angle: an angle that is greater than $180^\circ$ and less than $360^\circ$
$0^\circ < \theta < 360^\circ$$ - Full angle (complete rotation): an angle exactly equal to $360^\circ$
$$\theta = 360^\circ$$
Positive and negative degree measures
Angles in trigonometry are often drawn in the coordinate plane using standard position (covered in more detail elsewhere). In degrees:
- Positive angles are measured by rotating counterclockwise from the initial side.
- Negative angles are measured by rotating clockwise.
Examples:
- $60^\circ$ is a $60$-degree counterclockwise rotation.
- $-60^\circ$ is a $60$-degree clockwise rotation.
- $-90^\circ$ corresponds to rotating clockwise a quarter turn.
Even though degrees are traditionally between $0^\circ$ and $360^\circ$ in many geometric settings, in trigonometry you will often encounter angles larger than $360^\circ$ or less than $0^\circ$.
Coterminal angles in degrees
Two angles are coterminal if they share the same initial side and terminal side, even if they represent different amounts of rotation.
In degrees, you can add or subtract full turns ($360^\circ$) to get coterminal angles. If $\theta$ is any angle in degrees, then
$$
\theta + 360^\circ k
$$
is coterminal with $\theta$ for any integer $k$.
Examples:
- $30^\circ$, $390^\circ$, and $-330^\circ$ are all coterminal, because:
- $390^\circ = 30^\circ + 360^\circ$
- $-330^\circ = 30^\circ - 360^\circ$
- $-45^\circ$ is coterminal with $315^\circ$ because:
$$-45^\circ + 360^\circ = 315^\circ$$
Coterminal angles become especially useful when evaluating trigonometric functions for angles outside the basic range $0^\circ$ to $360^\circ$.
Quadrants and degree ranges
When angles are drawn in standard position, their terminal sides fall into one of four quadrants or onto the axes. In degrees, the quadrant is determined by the angle’s size:
- Quadrant I: between $0^\circ$ and $90^\circ$
$ ^\circ < \theta < 90^\circ$$
- Quadrant II: between $90^\circ$ and $180^\circ$
$^\circ < \theta < 180^\circ$$ - Quadrant III: between $180^\circ$ and $270^\circ$
$0^\circ < \theta < 270^\circ$$ - Quadrant IV: between $270^\circ$ and $360^\circ$
$0^\circ < \theta < 360^\circ$$
Angles exactly equal to $0^\circ$, $90^\circ$, $180^\circ$, $270^\circ$, or $360^\circ$ lie on the axes rather than inside a quadrant.
This quadrant information will later help you determine the signs of sine, cosine, and tangent for various angles.
Reducing degree measures to a standard range
In many problems, it is helpful to replace any angle by a coterminal angle between $0^\circ$ and $360^\circ$.
To reduce a positive angle $\theta$ in degrees:
- Subtract $360^\circ$ repeatedly until the result is between $0^\circ$ and $360^\circ$.
To reduce a negative angle:
- Add $360^\circ$ repeatedly until the result is between $0^\circ$ and $360^\circ$.
Examples:
- $\theta = 765^\circ$
Subtract 0^\circ$:
$5^\circ - 360^\circ = 405^\circ$$
Subtract 0^\circ$ again:
$5^\circ - 360^\circ = 45^\circ$$
So 5^\circ$ is coterminal with ^\circ$. - $\theta = -200^\circ$
Add 0^\circ$:
$$-200^\circ + 360^\circ = 160^\circ$$
So $-200^\circ$ is coterminal with 0^\circ$.
Degree subdivisions: minutes and seconds (optional detail)
Sometimes angles are measured more precisely using minutes and seconds, similar to time.
- $1^\circ = 60'$ (60 minutes)
- $1' = 60''$ (60 seconds)
Thus:
- $1^\circ = 60' = 3600''$
An angle may be written as:
$$
\theta = D^\circ M' S''
$$
where $D$ is degrees, $M$ is minutes, and $S$ is seconds.
Example:
- $30^\circ 15'$ means $30$ degrees and $15$ minutes.
- $45^\circ 30' 20''$ means $45$ degrees, $30$ minutes, $20$ seconds.
Converting between decimal degrees and $^\circ ' ''$
To convert from degrees–minutes–seconds to decimal degrees:
- Keep the degree part as is.
- Convert minutes to a fraction of a degree by dividing by $60$.
- Convert seconds to a fraction of a degree by dividing by $3600$.
- Add them.
For example:
$$
25^\circ 30' = 25^\circ + \frac{30}{60}^\circ = 25.5^\circ
$$
To convert from decimal degrees to degrees–minutes–seconds:
- Take the whole number part as degrees.
- Multiply the decimal part by $60$ to get minutes.
- Take the whole number part of that as minutes.
- Multiply the remaining decimal by $60$ to get seconds (often rounded).
These finer units are useful in fields like navigation, astronomy, and surveying, though in much of trigonometry you will mainly use decimal degrees.
Summary of degree facts to remember
- A full turn is $360^\circ$.
- Right angle: $90^\circ$, straight angle: $180^\circ$, full angle: $360^\circ$.
- Acute: $(0^\circ, 90^\circ)$, obtuse: $(90^\circ, 180^\circ)$, reflex: $(180^\circ, 360^\circ)$.
- You can add or subtract $360^\circ$ to find coterminal angles.
- Quadrant ranges:
- QI: $(0^\circ, 90^\circ)$
- QII: $(90^\circ, 180^\circ)$
- QIII: $(180^\circ, 270^\circ)$
- QIV: $(270^\circ, 360^\circ)$
- Optionally, degrees can be subdivided into minutes and seconds for more precision.
The separate chapter on radians will explain the radian unit and how to convert between degrees and radians.