The Central Idea
An angle measures the amount of rotation from one ray or direction to another.
Two common units are:
- degrees, based on \(360^\circ\) in one full turn;
- radians, based on the ratio of arc length to radius.
The two systems are linked by
\[ \boxed{180^\circ=\pi\text{ radians}}. \]Angles as Rotations
An angle can be understood dynamically as a rotation.
One ray begins in an initial position. A second ray marks the terminal position after rotation.
The amount and direction of that rotation determine the angle.
Positive and Negative Rotation
By standard mathematical convention:
- counterclockwise rotation is positive;
- clockwise rotation is negative.
Thus,
\[ 90^\circ \]represents a quarter-turn counterclockwise, while
\[ -90^\circ \]represents a quarter-turn clockwise.
A Full Rotation
One complete revolution returns a ray to its original direction.
In degrees,
\[ \boxed{1\text{ full turn}=360^\circ}. \]In radians,
\[ \boxed{1\text{ full turn}=2\pi}. \]Half-Turns and Quarter-Turns
A half-turn is
\[ 180^\circ=\pi. \]A quarter-turn is
\[ 90^\circ=\frac{\pi}{2}. \]Three quarter-turns give
\[ 270^\circ=\frac{3\pi}{2}. \]Degree Measure
The degree system divides a full revolution into \(360\) equal angular units.
One degree is therefore
\[ \boxed{ 1^\circ=\frac{1}{360}\text{ of a full turn} }. \]Degrees are convenient for geometry, navigation, surveying, and ordinary descriptions of direction.
Common Degree Angles
Several degree measures appear frequently:
\[ 30^\circ,\qquad 45^\circ,\qquad 60^\circ,\qquad 90^\circ,\qquad 180^\circ,\qquad 270^\circ,\qquad 360^\circ. \]These values correspond to useful fractions of a complete rotation.
Angle Classification by Degree Measure
For positive angles smaller than a full turn:
- acute: \(0^\circ<\theta<90^\circ\);
- right: \(\theta=90^\circ\);
- obtuse: \(90^\circ<\theta<180^\circ\);
- straight: \(\theta=180^\circ\);
- reflex: \(180^\circ<\theta<360^\circ\).
Why Radians Exist
Degrees are based on dividing a turn into \(360\) parts. Radians arise directly from the geometry of a circle.
A radian compares two lengths:
- the length of an arc;
- the radius of the circle.
This makes radian measure especially natural in formulas involving circles and change.
Definition of a Radian
Consider a circle of radius \(r\). An angle at the center intercepts an arc of length \(s\).
The angle in radians is
\[ \boxed{\theta=\frac{s}{r}}. \]One radian is the central angle that intercepts an arc whose length equals the radius.
Why Radians Are Dimensionless
In
\[ \theta=\frac{s}{r}, \]both \(s\) and \(r\) are lengths measured in the same unit.
Their units cancel:
\[ \frac{\text{length}}{\text{length}}=1. \]Radian measure is therefore dimensionless in the formal dimensional sense, even though the word radian is retained to identify the kind of quantity being measured.
Deriving a Full Turn in Radians
The circumference of a circle is
\[ C=2\pi r. \]For one full rotation, the intercepted arc is the entire circumference:
\[ s=2\pi r. \]Therefore,
\[ \theta=\frac{2\pi r}{r}=2\pi. \]Hence,
\[ \boxed{360^\circ=2\pi\text{ radians}}. \]The Fundamental Degree-Radian Equivalence
Dividing
\[ 360^\circ=2\pi \]by \(2\) gives
\[ \boxed{180^\circ=\pi}. \]This single equivalence is the basis of all degree-radian conversions.
Converting Degrees to Radians
Since
\[ 180^\circ=\pi, \]multiply a degree measure by
\[ \boxed{\frac{\pi}{180^\circ}}. \]Thus,
\[ \boxed{ \theta_{\text{rad}} = \theta_{\text{deg}} \frac{\pi}{180} }. \]Example: Convert \(60^\circ\) to Radians
\[ 60^\circ\cdot\frac{\pi}{180^\circ} = \frac{60\pi}{180}. \]Simplify:
\[ \boxed{\frac{\pi}{3}}. \]Example: Convert \(225^\circ\) to Radians
\[ 225^\circ\cdot\frac{\pi}{180^\circ} = \frac{225\pi}{180}. \]Divide numerator and denominator by \(45\):
\[ \boxed{\frac{5\pi}{4}}. \]Converting Radians to Degrees
To convert radians to degrees, multiply by
\[ \boxed{\frac{180^\circ}{\pi}}. \]Thus,
\[ \boxed{ \theta_{\text{deg}} = \theta_{\text{rad}} \frac{180}{\pi} }. \]Example: Convert \(\frac{2\pi}{3}\) to Degrees
\[ \frac{2\pi}{3} \cdot \frac{180^\circ}{\pi} = \frac{2(180^\circ)}{3}. \] \[ \boxed{120^\circ}. \]Example: Convert \(\frac{7\pi}{6}\) to Degrees
\[ \frac{7\pi}{6} \cdot \frac{180^\circ}{\pi} = 7(30^\circ). \] \[ \boxed{210^\circ}. \]A Useful Conversion Table
\[ \begin{array}{c|c} \text{Degrees} & \text{Radians}\\ \hline 0^\circ & 0\\ 30^\circ & \frac{\pi}{6}\\ 45^\circ & \frac{\pi}{4}\\ 60^\circ & \frac{\pi}{3}\\ 90^\circ & \frac{\pi}{2}\\ 120^\circ & \frac{2\pi}{3}\\ 135^\circ & \frac{3\pi}{4}\\ 150^\circ & \frac{5\pi}{6}\\ 180^\circ & \pi\\ 270^\circ & \frac{3\pi}{2}\\ 360^\circ & 2\pi \end{array} \]Standard Position
In the coordinate plane, an angle is in standard position when:
- its vertex is at the origin;
- its initial side lies on the positive \(x\)-axis.
The terminal side is determined by rotating through the angle.
Quadrants and Angles
For angles between \(0^\circ\) and \(360^\circ\):
- Quadrant I: \(0^\circ<\theta<90^\circ\);
- Quadrant II: \(90^\circ<\theta<180^\circ\);
- Quadrant III: \(180^\circ<\theta<270^\circ\);
- Quadrant IV: \(270^\circ<\theta<360^\circ\).
The same intervals can be written in radians.
Quadrants in Radians
- Quadrant I: \(0<\theta<\frac{\pi}{2}\);
- Quadrant II: \(\frac{\pi}{2}<\theta<\pi\);
- Quadrant III: \(\pi<\theta<\frac{3\pi}{2}\);
- Quadrant IV: \(\frac{3\pi}{2}<\theta<2\pi\).
Quadrantal Angles
An angle whose terminal side lies on a coordinate axis is called a quadrantal angle.
Common examples include
\[ 0^\circ,\quad 90^\circ,\quad 180^\circ,\quad 270^\circ,\quad 360^\circ, \]or equivalently,
\[ 0,\quad \frac{\pi}{2},\quad \pi,\quad \frac{3\pi}{2},\quad 2\pi. \]Coterminal Angles
Two angles are coterminal if they have the same initial and terminal sides but differ by one or more complete rotations.
In degrees,
\[ \boxed{\theta+360^\circ k} \]is coterminal with \(\theta\) for any integer \(k\).
In radians,
\[ \boxed{\theta+2\pi k}. \]Example: Coterminal Angles in Degrees
Starting with
\[ 45^\circ, \]add or subtract \(360^\circ\):
\[ 45^\circ+360^\circ=405^\circ, \] \[ 45^\circ-360^\circ=-315^\circ. \]Thus,
\[ \boxed{45^\circ,\ 405^\circ,\ -315^\circ} \]are coterminal.
Example: Coterminal Angles in Radians
Starting with
\[ \frac{\pi}{3}, \]add \(2\pi\):
\[ \frac{\pi}{3}+2\pi = \frac{\pi}{3}+\frac{6\pi}{3} = \frac{7\pi}{3}. \]Subtract \(2\pi\):
\[ \frac{\pi}{3}-2\pi = -\frac{5\pi}{3}. \]Angles Larger Than One Turn
An angle may exceed \(360^\circ\) or \(2\pi\).
For example,
\[ 810^\circ \]describes two complete counterclockwise turns plus another \(90^\circ\).
Its terminal side therefore matches the terminal side of
\[ 90^\circ. \]Negative Angles
Negative angles represent clockwise rotation.
For example,
\[ -\frac{\pi}{2} \]is a clockwise quarter-turn.
It is coterminal with
\[ -\frac{\pi}{2}+2\pi = \frac{3\pi}{2}. \]Radians and Arc Length
The defining relation
\[ \theta=\frac{s}{r} \]can be rearranged to give the arc-length formula
\[ \boxed{s=r\theta}, \]provided \(\theta\) is measured in radians.
Example: Arc Length
A circle has radius
\[ r=6\text{ cm}, \]and a central angle of
\[ \theta=\frac{\pi}{3}. \]Then
\[ s=r\theta = 6\left(\frac{\pi}{3}\right) = 2\pi. \] \[ \boxed{s=2\pi\text{ cm}}. \]Why the Arc-Length Formula Requires Radians
The formula
\[ s=r\theta \]follows directly from the radian definition
\[ \theta=\frac{s}{r}. \]If the angle is given in degrees, it must first be converted to radians.
Example: Arc Length from Degrees
A circle has radius \(10\text{ m}\) and central angle \(72^\circ\).
Convert the angle:
\[ 72^\circ\cdot\frac{\pi}{180^\circ} = \frac{2\pi}{5}. \]Then
\[ s=10\left(\frac{2\pi}{5}\right)=4\pi. \] \[ \boxed{s=4\pi\text{ m}}. \]Radians on the Unit Circle
On a unit circle,
\[ r=1. \]Therefore,
\[ s=r\theta=\theta. \]The numerical value of a radian angle equals the arc length it intercepts on the unit circle.
This is one reason radians are especially natural in trigonometry.
One Radian in Degrees
Convert \(1\) radian to degrees:
\[ 1\cdot\frac{180^\circ}{\pi} \approx57.2958^\circ. \]Therefore,
\[ \boxed{1\text{ rad}\approx57.3^\circ}. \]Degrees and Radians Measure the Same Quantity
Degree measure and radian measure are not different kinds of angles.
They are different numerical descriptions of the same rotation.
Just as
\[ 1\text{ m}=100\text{ cm}, \]the same angle can be written as
\[ 60^\circ=\frac{\pi}{3}. \]Exact Radian Values
When converting common degree measures to radians, keep \(\pi\) exact.
For example,
\[ 135^\circ=\frac{3\pi}{4} \]is more informative and exact than an immediate decimal approximation.
Decimal Radian Measures
Not every radian measure is written as a multiple of \(\pi\).
For example,
\[ 1.2\text{ radians} \]is a perfectly valid angle measure.
In degrees,
\[ 1.2\cdot\frac{180^\circ}{\pi} \approx68.75^\circ. \]Degrees, Minutes, and Seconds
Degree measure can be subdivided further.
One degree contains \(60\) arcminutes:
\[ \boxed{1^\circ=60'} \]and one arcminute contains \(60\) arcseconds:
\[ \boxed{1'=60''}. \]Therefore,
\[ 1^\circ=3600''. \]Example: Degrees-Minutes-Seconds to Decimal Degrees
Convert
\[ 32^\circ15'30'' \]to decimal degrees.
\[ 32+\frac{15}{60}+\frac{30}{3600}. \] \[ 32+0.25+0.008333\ldots = 32.258333\ldots \] \[ \boxed{32.2583^\circ\text{ approximately}}. \]Example: Decimal Degrees to Degrees-Minutes-Seconds
Convert
\[ 47.375^\circ. \]The whole-degree part is
\[ 47^\circ. \]Convert the decimal part:
\[ 0.375(60)=22.5'. \]Thus there are \(22'\) and
\[ 0.5(60)=30''. \]Therefore,
\[ \boxed{47^\circ22'30''}. \]Angular Speed
When an angle changes over time, the rate of change is called angular speed.
If an angle changes by \(\Delta\theta\) over a time interval \(\Delta t\), then average angular speed is
\[ \boxed{ \omega=\frac{\Delta\theta}{\Delta t} }. \]Common units include degrees per second, radians per second, and revolutions per minute.
Radians per Second
In scientific and engineering contexts, angular speed is often measured in radians per second:
\[ \text{rad/s}. \]Because one revolution equals \(2\pi\) radians, rotational rates can be converted naturally between revolutions and radians.
Example: Revolutions to Radians
A wheel rotates through \(3\) complete revolutions.
Since one revolution is \(2\pi\) radians,
\[ 3(2\pi)=6\pi. \]Therefore,
\[ \boxed{3\text{ revolutions}=6\pi\text{ radians}}. \]Angle Measure and Orientation
An angle can describe how one direction is oriented relative to another.
For example, perpendicular directions differ by
\[ 90^\circ \]or
\[ \frac{\pi}{2}. \]Opposite directions differ by
\[ 180^\circ=\pi. \]Angle Measure and Periodicity
Rotations repeat after one complete turn.
Therefore, angular quantities naturally exhibit periodicity:
\[ \theta \]and
\[ \theta+2\pi \]describe the same terminal direction in standard position.
This periodic structure becomes central in trigonometric functions.
Common Error: Mixing Degrees and Radians
Expressions involving angle measure must use a consistent unit.
For example,
\[ 60^\circ+\frac{\pi}{6} \]should not be combined numerically until one measure is converted.
Since
\[ \frac{\pi}{6}=30^\circ, \]the sum is
\[ \boxed{90^\circ}. \]Common Error: Using \(180\) Instead of \(360\) for a Full Turn
A straight angle is
\[ 180^\circ, \]but a complete revolution is
\[ 360^\circ. \]In radians, these correspond to
\[ \pi \]and
\[ 2\pi, \]respectively.
Common Error: Reversing the Conversion Factor
To convert degrees to radians, use
\[ \frac{\pi}{180}. \]To convert radians to degrees, use
\[ \frac{180}{\pi}. \]A useful check is that the unwanted unit should cancel.
Common Error: Using Degrees in \(s=r\theta\)
The formula
\[ s=r\theta \]requires \(\theta\) in radians.
If an angle is given as \(60^\circ\), first convert:
\[ 60^\circ=\frac{\pi}{3}. \]Only then should the angle be substituted into the arc-length formula.
Common Error: Ignoring Rotation Direction
The angles
\[ 90^\circ \]and
\[ -90^\circ \]have the same magnitude but represent opposite directions of rotation.
Signed angle measure preserves orientation information.
Common Error: Assuming Coterminal Means Equal
Coterminal angles have the same terminal side, but their numerical measures need not be equal.
For example,
\[ 30^\circ \]and
\[ 390^\circ \]are coterminal, but
\[ 30^\circ\ne390^\circ. \]They differ by one complete revolution.
A Reliable Procedure for Degree-to-Radian Conversion
- Write the degree measure.
- Multiply by \(\frac{\pi}{180^\circ}\).
- Cancel the degree unit.
- Reduce the numerical fraction.
- Keep \(\pi\) exact unless a decimal is required.
A Reliable Procedure for Radian-to-Degree Conversion
- Write the radian measure.
- Multiply by \(\frac{180^\circ}{\pi}\).
- Cancel \(\pi\) when possible.
- Simplify the numerical factor.
- Attach the degree symbol.
A Reliable Procedure for Arc Length
- Identify the radius \(r\).
- Identify the central angle \(\theta\).
- Convert \(\theta\) to radians if necessary.
- Use \(s=r\theta\).
- Keep exact values involving \(\pi\) when appropriate.
- Report the result in linear units.
Example: Full Degree-Radian Workflow
Convert
\[ 150^\circ \]to radians.
\[ 150^\circ\cdot\frac{\pi}{180^\circ} = \frac{150\pi}{180}. \]Reduce by \(30\):
\[ \boxed{\frac{5\pi}{6}}. \]Converting back,
\[ \frac{5\pi}{6}\cdot\frac{180^\circ}{\pi} = 150^\circ. \]The two conversions confirm one another.
Example: Full Arc-Length Workflow
A circle has radius
\[ 9\text{ cm} \]and central angle
\[ 140^\circ. \]Convert the angle:
\[ 140^\circ\cdot\frac{\pi}{180^\circ} = \frac{7\pi}{9}. \]Then
\[ s=9\left(\frac{7\pi}{9}\right)=7\pi. \] \[ \boxed{s=7\pi\text{ cm}}. \]Practice 1: Degrees to Radians
Convert
\[ 45^\circ \]to radians.
Practice 2: Degrees to Radians
Convert
\[ 300^\circ \]to radians.
Practice 3: Radians to Degrees
Convert
\[ \frac{5\pi}{3} \]to degrees.
Practice 4: Radians to Degrees
Convert
\[ -\frac{3\pi}{4} \]to degrees.
Practice 5: Coterminal Angle
Find one positive and one negative angle coterminal with
\[ 120^\circ. \]Practice 6: Coterminal Radian Angle
Find one positive angle greater than \(2\pi\) that is coterminal with
\[ \frac{3\pi}{4}. \]Practice 7: Arc Length
A circle has radius \(8\text{ cm}\) and central angle \(\frac{3\pi}{4}\). Find the arc length.
Practice 8: Arc Length from Degrees
A circle has radius \(12\text{ m}\) and central angle \(150^\circ\). Find the exact arc length.
Practice 9: Decimal Radians
Convert
\[ 2.5\text{ radians} \]to degrees, rounded to the nearest tenth.
Practice 10: Degrees-Minutes-Seconds
Convert
\[ 18^\circ30'0'' \]to decimal degrees.
Practice Answers
Practice 1
\[ 45^\circ\cdot\frac{\pi}{180^\circ} = \boxed{\frac{\pi}{4}}. \]Practice 2
\[ 300^\circ\cdot\frac{\pi}{180^\circ} = \frac{5\pi}{3}. \] \[ \boxed{\frac{5\pi}{3}}. \]Practice 3
\[ \frac{5\pi}{3}\cdot\frac{180^\circ}{\pi} = 300^\circ. \] \[ \boxed{300^\circ}. \]Practice 4
\[ -\frac{3\pi}{4}\cdot\frac{180^\circ}{\pi} = -135^\circ. \] \[ \boxed{-135^\circ}. \]Practice 5
Add and subtract one full turn:
\[ 120^\circ+360^\circ=480^\circ, \] \[ 120^\circ-360^\circ=-240^\circ. \] \[ \boxed{480^\circ\text{ and }-240^\circ}. \]Practice 6
\[ \frac{3\pi}{4}+2\pi = \frac{3\pi}{4}+\frac{8\pi}{4} = \frac{11\pi}{4}. \] \[ \boxed{\frac{11\pi}{4}}. \]Practice 7
\[ s=r\theta = 8\left(\frac{3\pi}{4}\right) = 6\pi. \] \[ \boxed{s=6\pi\text{ cm}}. \]Practice 8
Convert the angle:
\[ 150^\circ=\frac{5\pi}{6}. \]Then
\[ s=12\left(\frac{5\pi}{6}\right)=10\pi. \] \[ \boxed{s=10\pi\text{ m}}. \]Practice 9
\[ 2.5\cdot\frac{180^\circ}{\pi} \approx143.2^\circ. \] \[ \boxed{143.2^\circ}. \]Practice 10
\[ 18+\frac{30}{60} = 18.5. \] \[ \boxed{18.5^\circ}. \]What to Remember
- An angle measures rotation or directional change.
- Counterclockwise rotation is positive and clockwise rotation is negative.
- One full turn is \(360^\circ=2\pi\) radians.
- \(180^\circ=\pi\) radians.
- Degrees convert to radians by multiplying by \(\frac{\pi}{180}\).
- Radians convert to degrees by multiplying by \(\frac{180}{\pi}\).
- Radian measure is defined by \(\theta=\frac{s}{r}\).
- Arc length is \(s=r\theta\) when \(\theta\) is in radians.
- Coterminal angles differ by complete rotations.
- Radians provide the natural angular language for circles, trigonometry, and calculus.
The central connection is rotation can be measured either by dividing a turn into degrees or by comparing arc length with radius.
Where We Go Next
Angular measure adds rotation to the measurement of spatial relationships. The next lesson, Units, Derived Units, and Dimensional Consistency, broadens the measurement framework by examining how units are constructed, combined, converted, and checked across mathematical and applied contexts.
That framework will connect length, angle, area, volume, rate, and other quantities through dimensional structure.