Radian measure defined as arc length divided by radius, and why that ratio is the same for every circle: one radian is the angle cutting off an arc as long as the radius, a full revolution is two pi radians, and because a length over a length has no units, an angle in radians is a pure real number. Includes both conversion factors, standard position and coterminal angles restated in radians, and the wrapping of the number line onto the unit circle that identifies each real number with an angle.
Subject: Trigonometry · 65 slides · symbolic lesson
Open the interactive version of this deck
Title
Trigonometry · Chapter 10 — Foundations of Trigonometry
§10.1 Angles and their Measure, pp. 700-706
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 700-706 — the pages these objectives are drawn from
Warm-up
You met pi in geometry as the number that turns a diameter into a circumference. That single fact is about to become a unit of angle.
Discussion prompt
Take any two circles, one tiny and one enormous. In each, measure the circumference and divide by the radius. What do you get, and why is it the same for both?
Hint: Start from the definition of pi as circumference over diameter, and remember how the diameter relates to the radius.
Answer:
\[ \pi = \frac{C}{d} \qquad d = 2r \quad\Longrightarrow\quad \frac{C}{r} = 2\pi \]
Both circles give two pi, and it is the same for both because all circles are similar — scaling a circle multiplies its circumference and its radius by the same factor, so their ratio never moves. That single unchanging ratio is what this lesson turns into a unit of angle.
Stitz and Zeager point out that this is quietly a theorem rather than a definition, and a far from obvious one. We will use it as given, as geometry does.
Concept
Degrees answer the question with a number somebody chose. Radians answer it with a number the circle supplies: take the arc the angle cuts off, and measure it in radius-lengths. That count is the radian measure.
radian measure — For a central angle subtending an arc of length s in a circle of radius r, the radian measure of the angle is s divided by r. It counts how many radius-lengths of arc the angle sweeps out.
\[ \theta = \frac{s}{r} \]
Because the ratio of arc to radius is the same in every circle for a given angle, this measures the angle and not the circle — which is the only reason it is allowed to be called a measure of the angle at all.
Figure (svg): A circle with radius r, an arc of length s, and the central angle theta subtended by that arc, with the defining ratio theta equals s over r written beside it
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 701-701
Section
Section 1
Concept
Read the definition as an instruction. If the radian measure is 1 then the arc equals the radius. If it is 2 the arc is two radii long, if it is 3 it is three, and so on. Radian measure is a count.
Notice that nothing here was chosen. Two pi appeared because that is how many radius-lengths fit around a circle, which is a fact about circles rather than a convention about notation.
Figure (svg): A circle with a central angle whose subtended arc has been laid out to exactly the same length as the radius, so the angle measures one radian, next to a second circle where four radius-lengths of arc give an angle of four radians
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 701-702
Picture it
Radians are harder to compute with by hand. It is worth being clear about what you get in exchange.
Figure (svg): Two columns contrasting what degree measure is with what radian measure is, showing that degrees are a chosen convention while radians are forced by the circle
You are trading convenient numbers for a measure that is dimensionless and tied to arc length. Every formula from Lesson 10.1c onward, and every derivative in calculus, cashes in that trade.
Worked example
Exactly the move you used for degrees, with a different constant.
\[ \text{Find the radian measure of half a revolution, a quarter revolution, and one twelfth of a revolution.} \]
Write down what one revolution measures in radians
Why: The circumference is two pi radii, so a full turn sweeps two pi radius-lengths of arc.
\[ 1\text{ revolution } = 2 \pi \]
Take half of it
Why: Measuring proportionately works the same way it did in degrees.
\[ (\frac{1}{2}) (2 \pi) = \pi \]
Take a quarter of it
Why: Same move again.
\[ (\frac{1}{4}) (2 \pi) = \frac{\pi}{2} \]
Take one twelfth of it
Why: Two pi over twelve reduces to pi over six.
\[ (\frac{1}{12}) (2 \pi) = \frac{\pi}{6} \]
Figure (svg): The solution to Worked example radian measure from a fraction of a turn shown as a ladder of expressions, one row per legal move
\[ \tfrac{1}{2}(2\pi) = \pi \qquad \tfrac{1}{4}(2\pi) = \tfrac{\pi}{2} \qquad \tfrac{1}{12}(2\pi) = \tfrac{\pi}{6} \]
Verify: check against the degree answers
Why: Half a revolution was 180 degrees and is pi radians; a quarter was 90 degrees and is pi over two; one twelfth was 30 degrees and is pi over six. The same three angles, relabelled.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 702-702
Prediction
An angle has radian measure 3.
Predict first
Is the terminal side in quadrant one, two, three or four?
Correct: Quadrant II.
Why: Pi is about 3.14, so a measure of 3 is just under pi, which is half a revolution. An angle a little short of half a turn has swept past the positive y-axis at pi over two, or about 1.57, but has not yet reached the negative x-axis at 3.14. That puts the terminal side in quadrant two, close to the negative x-axis. Comparing a decimal radian measure against the landmarks 1.57, 3.14 and 4.71 is the fastest way to place it.
Worked example
This direction is the one that places a terminal side without any conversion to degrees at all.
\[ \text{What fraction of a revolution is } \tfrac{9\pi}{4} \text{? And } -\tfrac{4\pi}{3} \text{?} \]
Divide the measure by two pi
Why: Dividing by the size of one revolution counts how many revolutions the measure represents.
\[ \frac{9 \pi / 4}{2 \pi} \]
Cancel the pi
Why: Pi appears in both numerator and denominator, which is the usual reason radian arithmetic is easier than it looks.
\[ \frac{9}{8} \]
Do the same for the second measure, keeping the sign
Why: The negative sign records clockwise rotation and survives the division untouched.
\[ \frac{4 \pi / 3}{2 \pi} = \frac{2}{3}\text{ clockwise} \]
Interpret each result
Why: Nine eighths is one whole revolution plus an eighth; two thirds of a turn clockwise passes the negative y-axis.
\[ \text{one turn plus } \frac{1}{8};\text{ and } \frac{2}{3}\text{ turn clockwise} \]
Figure (svg): The solution to Worked example what fraction of a turn is this shown as a ladder of expressions, one row per legal move
\[ \frac{9\pi/4}{2\pi} = \tfrac{9}{8} \qquad \frac{-4\pi/3}{2\pi} = -\tfrac{2}{3} \]
Verify: place the terminal sides
Why: An eighth of a turn past the start lands midway through quadrant one, so nine pi over four is a quadrant one angle. Two thirds of a turn clockwise passes 180 degrees clockwise at half a turn and continues, landing in quadrant two. Both match the book's classifications.
Trap
A student reasons: a big circle has a longer arc for the same angle, so the same angle must have a bigger radian measure in a big circle.
\[ \text{big circle: } s = 20, \; r = 10 \qquad \text{small circle: } s = 2, \; r = 1 \]
Compare the arc lengths and conclude the measures differ
Why: Only the numerator was looked at. The definition is a ratio, and the denominator changed too.
\[ \text{big circle: } \theta = \tfrac{20}{10} = 2 \qquad \text{small circle: } \theta = \tfrac{2}{1} = 2 \]
Divide arc by radius in each circle separately
Why: Scaling a circle multiplies s and r by the same factor, so the quotient is unchanged.
Both give two radians. The arc did get twenty times longer, but so did the radius, and the ratio cannot tell the difference. That invariance is precisely what makes the ratio a legitimate measure of the angle rather than a measurement of the circle.
Fill the middle
What fraction of a revolution is five pi over three?
Fill in the blanks
\frac1/(2 pi)5/6 = \frac______ \cdot ___ = ___
Why: Dividing by two pi is multiplying by its reciprocal, and the pi cancels immediately, leaving five over six. Five sixths of a revolution is 300 degrees, which lands in quadrant four — and knowing it is five sixths of a turn places the terminal side just as well as knowing it is 300 degrees does.
Estimation
One radian is being converted to degrees.
Predict first
Roughly how big is one radian in degrees?
Correct: About 57 degrees.
Why: Half a revolution is pi radians and also 180 degrees, so one radian is 180 divided by pi, which is about 57.2958 degrees. A useful way to feel this: since pi is a bit more than 3, a radian must be a bit less than a third of 180. It is worth carrying this number, because it tells you a radian is a large angle — a little under two thirds of a right angle — and that catches sign and scale errors quickly.
Socratic
Degrees carry a symbol. Radians are written as bare numbers, so pi over three is an angle and also just a number.
Discussion prompt
Explain why radian measure is dimensionless, and say what practical consequence that has for the formula relating arc length to angle.
Hint: Look at the definition and ask what units the numerator and denominator carry.
Answer:
\[ \theta = \frac{s}{r} = \frac{\text{length}}{\text{length}} \]
Both s and r are lengths, so the units cancel and nothing is left. Radian measure is a pure number, which is why the book says we use the word radians only when we need to and never a symbol.
The practical consequence is enormous. Rearranging gives s equals r theta with no conversion factor of any kind, and in Lesson 10.1c the velocity formula comes out as v equals r omega for the same reason. In degrees both formulas would need a factor of pi over 180 bolted on, and every calculus formula for a trigonometric derivative would carry it too.
Section
Section 2
Concept
One revolution counterclockwise measures 360 degrees, and the same angle measures two pi radians. Everything about conversion follows from setting those equal and reducing.
\[ 360^\circ = 2\pi \text{ radians} \quad\Longleftrightarrow\quad 180^\circ = \pi \text{ radians} \]
That gives a ratio equal to one, and you may multiply by it freely. Which way up you write it is decided by the unit you want to cancel, exactly as in any other unit conversion.
Figure (svg): The two conversion factors between degrees and radians, shown as fractions with the unit that cancels highlighted in each direction
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 704-704
Picture it
You do not need to memorise these now. You will absorb them by using them.
Figure (svg): A unit circle with the sixteen standard angles marked and labelled in radians, from zero through pi over six, pi over four and pi over three all the way round to eleven pi over six
Look at the denominators rather than the whole fractions. Everything on this circle is a multiple of pi over six, pi over four, or pi over three, and those three come from the two special triangles you already know.
Worked example
Convert 60 degrees to radians, exactly, in terms of pi.
\[ \text{Convert } 60^\circ \text{ to radian measure.} \]
Choose the factor that cancels degrees
Why: Degrees are upstairs in the given measure, so the factor must have degrees downstairs.
\[ \text{multiply by } \pi / 180 ^\circ \]
Write the multiplication with units
Why: Writing the units is what makes the choice of factor self-checking.
\[ 60 ^\circ \times \pi / (180 ^\circ) \]
Cancel and reduce
Why: Sixty over 180 reduces to one third.
\[ 60 \pi / 180 = \frac{\pi}{3} \]
Leave the answer in terms of pi
Why: The exact form is more useful than a decimal, and every later lesson expects it.
\[ \frac{\pi}{3} \]
Figure (svg): The solution to Worked example degrees to radians shown as a ladder of expressions, one row per legal move
\[ 60^\circ \cdot \frac{\pi \text{ radians}}{180^\circ} = \frac{\pi}{3} \]
Verify: convert back
Why: Pi over three times 180 over pi cancels the pi and gives 180 over 3, which is 60 degrees. The round trip returns the original measure.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 704-704
Matching
These six are worth recognising on sight by the end of the chapter.
Match the pairs
Why: Each one is the degree measure divided by 180, with the result written as a multiple of pi. Thirty over 180 is one sixth, 45 over 180 is one quarter, 120 over 180 is two thirds, 270 over 180 is three halves, and 225 over 180 is five quarters. Reading the fraction as a fraction of a half-turn is faster than the formal multiplication once you trust it.
Worked example
Convert negative five pi over six to degrees. The sign rides along untouched.
\[ \text{Convert } -\frac{5\pi}{6} \text{ to degree measure.} \]
Choose the factor that cancels radians
Why: Radians are upstairs, so the factor needs radians downstairs and degrees upstairs.
\[ \text{multiply by } 180 ^\circ / \pi \]
Write the multiplication and cancel the pi
Why: The pi appears top and bottom and disappears immediately.
\[ -5(180) / 6 \]
Do the arithmetic
Why: One hundred eighty divided by six is thirty, times five is 150.
\[ -150 \]
Keep the negative sign
Why: A negative radian measure is a clockwise rotation and stays clockwise after conversion.
\[ -150 ^\circ \]
Figure (svg): The solution to Worked example radians to degrees, with a negative shown as a ladder of expressions, one row per legal move
\[ -\frac{5\pi}{6} \cdot \frac{180^\circ}{\pi \text{ radians}} = -150^\circ \]
Verify: check the quadrant both ways
Why: Negative 150 degrees sweeps clockwise past the negative y-axis at negative 90 and stops 60 degrees short of the negative x-axis, landing in quadrant three. In radians, negative five pi over six is five sixths of pi clockwise, which is five sixths of a half-turn — again just short of the negative x-axis, in quadrant three. The two agree.
Error analysis
A student converts pi over four radians to degrees.
Annotate
On: \( \frac{\pi}{4} \cdot \frac{\pi}{180} = \frac{\pi^2}{720} \approx 0.0137 \)
Write the units into the fraction every time. A factor with the wrong unit downstairs is visible instantly, whereas a bare number gives you nothing to check against.
Faded example
Convert 135 degrees to radians.
Fill in the blanks
135^\circ \cdot \frac180135} = \frac3\pi}___ = \frac___\pi}___
Why: The factor pi over 180 cancels the degrees. Then 135 over 180 reduces: both divide by 45, giving 3 over 4. So 135 degrees is three pi over four radians, which is three eighths of a revolution and lands midway through quadrant two — the same place 135 degrees does.
Discrimination
Some of these pairs describe one angle written two ways, and some do not.
Sort into buckets
Sort each pair by whether the two measures are the same angle.
Edge cases
Every conversion in this lesson has produced a nice multiple of pi, because every angle chosen was a nice fraction of a revolution.
Discussion prompt
What is 1 degree in radians, and what does the answer tell you about why the two systems never line up on whole numbers except at zero?
Hint: Think about what kind of number pi is.
Answer:
\[ 1^\circ = \frac{\pi}{180} \approx 0.01745 \]
One degree is pi over 180 radians, which is irrational, and one radian is 180 over pi degrees, which is also irrational. So apart from zero, no angle has a whole number of degrees and a whole number of radians at the same time.
That is not a defect; it is the reason radian answers are left in terms of pi. Converting pi over three to 1.047 throws away exactness for no gain, and every later lesson expects the exact form.
Section
Section 3
Concept
Every idea from the previous lesson survives the change of unit untouched. Standard position is still vertex at the origin and initial side on the positive x-axis, positive is still counterclockwise, and coterminal still means sharing a terminal side. Only the size of one revolution changed.
\[ \theta \text{ is coterminal with } \alpha \iff \theta = \alpha + 2\pi k, \quad k \in \mathbb{Z} \]
The trick that makes the arithmetic painless is to rewrite two pi with the same denominator as the angle you are adding it to. To work with pi over six, write two pi as twelve pi over six.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 702-702
Picture it
This is the unit circle you saw a moment ago. Every label on it is now doing double duty as an angle and as a real number.
Figure (svg): A unit circle with the sixteen standard angles marked and labelled in radians, from zero through pi over six, pi over four and pi over three all the way round to eleven pi over six
By Lesson 10.2b you will read coordinates off this circle. For now it is enough to place a terminal side from a radian measure without translating to degrees first.
Worked example
Example 10.1.3, part 1. Graph pi over six in standard position and give three coterminal angles.
\[ \alpha = \frac{\pi}{6} \]
Find what fraction of a revolution this is
Why: Divide by two pi; the pi cancels and the fraction is what places the terminal side.
\[ \frac{\frac{\pi}{6}}{2 \pi} = \frac{1}{12} \]
Rotate one twelfth of a revolution counterclockwise
Why: The measure is positive, so the sweep is counterclockwise. One twelfth of a turn is a short sweep.
Rewrite two pi with denominator six
Why: This is the arithmetic trick: matching denominators before adding avoids every common-denominator slip.
\[ 2 \pi = 12 \pi / 6 \]
Add and subtract multiples of twelve pi over six
Why: Taking k equal to 1, then negative 1, then 2 gives three coterminal angles.
\[ 13 \pi / 6, -11 \pi / 6, 25 \pi / 6 \]
Figure (svg): The angle pi over six drawn in standard position, a short counterclockwise sweep ending in the first quadrant one twelfth of the way around the circle
\[ \frac{\pi}{6} \in \text{QI} \qquad \theta = \frac{\pi}{6} + \frac{12\pi}{6}k \;\Longrightarrow\; \frac{13\pi}{6}, \; -\frac{11\pi}{6}, \; \frac{25\pi}{6} \]
Verify: check each difference
Why: Thirteen pi over six minus pi over six is twelve pi over six, which is two pi. Pi over six minus negative eleven pi over six is also twelve pi over six. And twenty-five pi over six minus pi over six is twenty-four pi over six, which is four pi, two full revolutions. Every difference is a whole multiple of two pi.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 702-703
Sorting
Place each one by its fraction of a revolution. Do not convert to degrees.
Sort into buckets
Sort each radian measure by where its terminal side lies.
Worked example
Example 10.1.3, part 2. Graph negative four pi over three and classify it.
\[ \beta = -\frac{4\pi}{3} \]
Find the fraction of a revolution
Why: Ignore the sign for the size, then put it back for the direction.
\[ \frac{4 \pi / 3}{2 \pi} = \frac{2}{3} \]
Rotate two thirds of a revolution clockwise
Why: Negative means clockwise. Half a turn clockwise reaches the negative x-axis; two thirds goes a sixth of a turn further.
\[ \text{past } 180 ^\circ\text{ clockwise by } 60 ^\circ \]
Read off the quadrant
Why: A sixth of a turn past the negative x-axis going clockwise puts the ray above the axis, in quadrant two.
Find coterminal angles using six pi over three
Why: Rewrite two pi with denominator three to keep the arithmetic clean.
\[ 2 \pi = 6 \pi / 3 \]
Figure (svg): The solution to Worked example a negative measure past a half turn shown as a ladder of expressions, one row per legal move
\[ -\frac{4\pi}{3} \in \text{QII} \qquad \theta = -\frac{4\pi}{3} + \frac{6\pi}{3}k \;\Longrightarrow\; \frac{2\pi}{3}, \; -\frac{10\pi}{3} \]
Verify: cross-check in degrees
Why: Negative four pi over three converts to negative 240 degrees, and adding 360 gives 120 degrees, which lies between 90 and 180 — quadrant two. The radian reasoning and the degree reasoning agree.
Trap
\[ \text{An angle coterminal with } \frac{\pi}{4} \text{ is } \frac{\pi}{4} + \pi = \frac{5\pi}{4}. \]
Add pi, on the grounds that pi is the natural constant here
Why: In radians pi looks like the fundamental unit, so it gets added by reflex. But pi is only HALF a revolution.
Pi over four is in quadrant one and five pi over four is in quadrant three, so the terminal sides point in opposite directions.
\[ \text{An angle coterminal with } \frac{\pi}{4} \text{ is } \frac{\pi}{4} + 2\pi = \frac{9\pi}{4}. \]
Add two pi, a whole revolution
Why: Rewrite two pi as eight pi over four first, so the addition is a single numerator sum.
\[ \frac{\pi}{4} + \frac{8\pi}{4} = \frac{9\pi}{4} \]
This is the exact radian twin of adding 180 instead of 360 in the previous lesson, and it has the same tell: the quadrant changed. Coterminal angles never change quadrant.
Fill the middle
Find the angle coterminal with nineteen pi over four that lies between zero and two pi.
Fill in the blanks
\frac216 - 3 \cdot 2\pi = \frac______ - \frac___\pi}___ = \frac___\pi}___
Why: Nineteen pi over four divided by two pi is nineteen eighths, which is two and three eighths, so two whole revolutions come out. Two revolutions is four pi, written as sixteen pi over four to match denominators. Nineteen minus sixteen leaves three pi over four, which is in the required window and lands in quadrant two.
Ranking
Mixed notation on purpose. Put everything into one form before comparing.
Put in order
Why: Convert everything to degrees first: pi over two is 90, two pi over three is 120, and 2 radians is 2 times 57.3, which is about 114.6. Sorting the five values 90, 100, 114.6, 120 and 150 gives pi over two, then 100 degrees, then 2 radians, then two pi over three, then 150 degrees. The bare radian is the trap: 2 radians looks small beside two pi over three but is in fact larger than 100 degrees and smaller than 120, and nothing about the notation tells you that until you convert.
Explain it to yourself
The coterminal formula changed from adding 360 k to adding two pi k.
Discussion prompt
Explain why that is the only change needed, and what would have gone wrong if the formula had kept the number 360 while the angles switched to radians.
Hint: What does the constant in that formula actually represent?
Answer:
The constant in the coterminal formula is not the number 360 or the number two pi as such — it is one full revolution, expressed in whatever unit the angles are being measured in. Change the unit and you must change the numeral, because the thing it names has not changed at all.
Keeping 360 while measuring in radians would mean adding about 57 revolutions each time k stepped by one. The terminal side would still land in the same place, by accident, since 360 is a whole number of revolutions in radians only if it were a multiple of two pi — and it is not. So the formula would simply be wrong: pi over six plus 360 is not coterminal with pi over six.
Section
Section 4
Concept
On the unit circle the radius is one, so the defining ratio becomes the arc length divided by one, which is the arc length itself. The radian measure of the angle and the length of the arc it cuts off are the same number.
wrapping function — The association of each real number t with an oriented arc on the unit circle beginning at the point with coordinates one and zero, of length the absolute value of t, running counterclockwise when t is positive and clockwise when t is negative.
\[ \text{On the Unit Circle } x^2 + y^2 = 1: \quad \theta = \frac{s}{r} = \frac{s}{1} = s \]
So every real number names an angle, and every angle names a real number. That identification is what lets sine and cosine be functions of a real variable in the next section, rather than functions of a geometric object.
Figure (svg): The real number line drawn straight, then shown wrapped counterclockwise around a unit circle so that each real number t corresponds to an arc of length t starting from the point one comma zero
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 704-705
Picture it
Picture the number line as a thread and the unit circle as a spool. Positive numbers wind counterclockwise from the point one comma zero; negative numbers wind clockwise from the same place.
Figure (svg): The real number line drawn straight, then shown wrapped counterclockwise around a unit circle so that each real number t corresponds to an arc of length t starting from the point one comma zero
Nothing is lost in the winding except uniqueness: infinitely many real numbers land on the same point, and those numbers are exactly the coterminal ones.
Worked example
Example 10.1.4, parts 1 and 2. Sketch the oriented arc on the unit circle for each value of t.
\[ t = \frac{3\pi}{4} \qquad t = -2\pi \]
Find the fraction of a revolution for the first value
Why: Three pi over four divided by two pi is three eighths.
\[ \frac{3}{8}\text{ of } a\text{ revolution} \]
Wind counterclockwise from the point one comma zero
Why: Three eighths of a turn passes the quarter mark and stops midway through quadrant two.
Handle the second value
Why: Negative two pi is exactly one revolution, and the sign makes it clockwise.
Say where the second arc ends
Why: A full revolution returns to the starting point, so the arc ends where it began even though it has length two pi.
\[ \text{arc ends at } (1, 0) \]
Figure (svg): The solution to Worked example sketch the arc for a real number shown as a ladder of expressions, one row per legal move
\[ t = \tfrac{3\pi}{4}: \text{ QII} \qquad t = -2\pi: \text{ back to } (1,0) \]
Verify: check the arc lengths
Why: The first arc should have length three pi over four, about 2.36, and the circumference of the unit circle is two pi, about 6.28. Three eighths of 6.28 is 2.36, matching. The second arc has length two pi, exactly one circumference, which is why it closes.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 705-705
Translation
Each value of t wraps to a point on the unit circle. Say where it lands.
Match the pairs
Why: Compare each value against the landmarks. Pi is exactly half a revolution, landing on the negative x-axis. One is less than 1.57, so it is short of a quarter turn and stays in quadrant one. Four is between 3.14 and 4.71, so it is past the half turn and in quadrant three. Negative pi over two is a quarter turn clockwise, landing on the negative y-axis.
Worked example
Example 10.1.4, parts 3 and 4. This is the case that shows the identification is with ALL real numbers, not just the tidy ones.
\[ t = -2 \qquad t = 117 \]
Locate negative two against the landmarks
Why: Pi over two is about 1.57 and pi is about 3.14, so two lies between them.
\[ 1.57 < 2 < 3.14 \]
Wind two units clockwise
Why: Clockwise, 1.57 units reaches the negative y-axis and two units goes a little past it.
Count the revolutions in 117
Why: Divide by two pi to see how many complete windings happen before the endpoint.
\[ \frac{117}{2 \pi} = 18.62 \]
Interpret the leftover
Why: Eighteen complete revolutions, then 0.62 of a revolution, which is just short of five eighths.
Figure (svg): The solution to Worked example a real number that is not a multiple of pi shown as a ladder of expressions, one row per legal move
\[ -2: \text{ just past } -\tfrac{\pi}{2} \text{ into QIII} \qquad 117 = 18(2\pi) + 0.62(2\pi) \]
Verify: check the leftover as a fraction
Why: Eighteen revolutions is 18 times 6.283, which is 113.1. Subtracting from 117 leaves 3.90 radians. Since pi is 3.14 and three pi over two is 4.71, a value of 3.90 sits between them, which is quadrant three — matching the answer.
Error analysis
A student is asked where the real number t equal to 5 lands on the unit circle.
Annotate
On: \( t = 5 \;\longrightarrow\; 5 \text{ revolutions} \;\longrightarrow\; \text{back at } (1,0) \)
The four landmark values 1.57, 3.14, 4.71 and 6.28 are worth carrying in your head. Any bare radian measure can be placed in seconds by comparing against them, with no conversion at all.
Prediction
Two real numbers, t equal to 3 and t equal to 3 plus two pi, are wrapped onto the unit circle.
Predict first
What is true of the two endpoints?
Correct: They are the same point.
Why: Adding two pi to a real number adds exactly one full circumference to the arc, which winds the thread once more around the spool and returns to the same place. In angle language the two numbers are coterminal. This is the geometric reason sine and cosine will be periodic with period two pi, and it is worth seeing before those functions are defined rather than after.
Invariant
Follow the endpoint as t increases by two pi each step, starting from t equal to pi over three.
Step through it
The value of t has grown from about 1.05 to about 19.90. What has not changed, and what has?
The endpoint on the circle is invariant; the arc length is not. Every quantity that will be defined from the endpoint — the coordinates, and therefore cosine and sine — inherits that invariance and becomes periodic. Every quantity defined from the arc length, such as total distance travelled, does not.
Missing information
A problem says: a real number t wraps to a point in quadrant two on the unit circle. Find t.
Discussion prompt
Why can this not be answered as asked? Name two different things you could be told that would each make the answer unique.
Hint: How many real numbers wrap into quadrant two?
Answer:
Infinitely many. Every t strictly between pi over two and pi lands in quadrant two, and so does each of those plus any whole multiple of two pi, in both directions. Quadrant two is a region, and the wrap sends infinitely many numbers into it.
Either of these fixes it. Give the exact endpoint — say, the point with coordinates negative root two over two and root two over two — and restrict t to a single revolution such as the interval from zero to two pi; that names one number, three pi over four. Or give one value of t and ask for the unique coterminal value in a stated window.
From Lesson 10.2b onward every 'find all angles' question comes with a window attached, and this is the job the window is doing.
Section
Section 5
Concept
Converting to degrees to place a terminal side works, but it is slow and it is a crutch. The faster route is to read the radian measure directly as a fraction of a revolution and place the ray from that.
Doing this fluently is the difference between a chapter that feels like translation and one that feels like reading. Every remaining lesson in Chapter 10 states its angles in radians and expects you to see them.
| Radian measure | Fraction of a revolution | Where the ray points |
|---|---|---|
| pi over 6 | one twelfth | Quadrant I, near the x-axis |
| pi over 2 | one quarter | Straight up, quadrantal |
| two pi over 3 | one third | Quadrant II |
| pi | one half | Straight left, quadrantal |
| five pi over 4 | five eighths | Quadrant III, midway |
| seven pi over 4 | seven eighths | Quadrant IV, midway |
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 702-703
Picture it
A common mistake is to imagine a radian as small, because pi over six and pi over four look like small fractions.
Figure (svg): An angle of one radian drawn in standard position beside a right angle of pi over two radians, showing that one radian is a little under two thirds of a right angle
Carry the number 57.3 degrees. It stops you reading a measure of 4 as a small angle when it is actually most of a half-turn plus a bit.
Worked example
Do not translate to degrees. Read the fraction and rotate.
\[ \text{Place the terminal side of } \theta = \frac{11\pi}{6} \text{ and classify it.} \]
Divide by two pi
Why: Write two pi as twelve pi over six so the division is a numerator comparison.
\[ \frac{11 \pi / 6}{12 \pi / 6} = \frac{11}{12} \]
Read the fraction
Why: Eleven twelfths of a revolution is one twelfth short of a full turn.
\[ \frac{1}{12}\text{ short of } a\text{ full turn} \]
Rotate that much counterclockwise
Why: Almost all the way round, stopping just below the positive x-axis.
Classify
Why: Between three pi over two and two pi is quadrant four.
Figure (svg): The solution to Worked example place a terminal side with no conversion shown as a ladder of expressions, one row per legal move
\[ \frac{11\pi}{6} = \tfrac{11}{12} \text{ of a revolution} \;\Longrightarrow\; \text{QIV} \]
Verify: cross-check in degrees
Why: Eleven pi over six converts to 330 degrees, which is between 270 and 360, so quadrant four. The fraction route and the conversion route agree, but the fraction route needed no arithmetic beyond cancelling.
Elimination
Rule out three, and say where each of them actually lands.
Eliminate the wrong options
Which angle has its terminal side in quadrant three?
Survives elimination: B
Why: Seven pi over six is seven twelfths of a revolution, which is one twelfth past the half turn. Half a turn ends on the negative x-axis, so one twelfth more puts the ray just below it, in quadrant three. All four options share the denominator six on purpose: the numerator alone decides the quadrant, with 1 to 5 giving quadrants one and two, 7 to 11 giving quadrants three and four, and 6 landing on the axis.
Worked example
Anything past two pi is reduced before it is placed. This is the single most common opening move in Chapter 10.
\[ \text{Place the terminal side of } \theta = \frac{29\pi}{6}. \]
Compare with two pi written as twelve pi over six
Why: Twenty-nine sixths is more than twelve sixths, so at least one revolution comes out.
\[ 29 > 12,\text{ so reduce} \]
Subtract one revolution
Why: Twelve pi over six is two pi.
\[ 29 \pi / 6 - 12 \pi / 6 = 17 \pi / 6 \]
Subtract another
Why: Seventeen is still bigger than twelve.
\[ 17 \pi / 6 - 12 \pi / 6 = 5 \pi / 6 \]
Place the reduced measure
Why: Five twelfths of a revolution is just short of half a turn.
Figure (svg): The solution to Worked example reduce a large measure first shown as a ladder of expressions, one row per legal move
\[ \frac{29\pi}{6} - 2(2\pi) = \frac{29\pi}{6} - \frac{24\pi}{6} = \frac{5\pi}{6} \in \text{QII} \]
Verify: count the revolutions independently
Why: Twenty-nine sixths divided by two, which is twelve sixths, gives 29 over 12, about 2.42. So two whole revolutions come out and 0.42 of one remains, and 0.42 of a turn is just past the quarter mark — quadrant two, as found.
Trap
\[ \text{What fraction of a revolution is } \frac{3\pi}{4}\text{?} \]
Divide by pi
Why: Pi is the constant in sight, so it becomes the thing divided by. But pi is a half revolution, not a whole one.
\[ \frac{3\pi/4}{\pi} = \frac{3}{4} \]
This says three quarters of a revolution, which would land in quadrant three. The angle is actually in quadrant two, so the answer is not merely mislabelled — it puts the ray in the wrong place.
\[ \frac{3\pi/4}{2\pi} = \frac{3}{8} \]
Divide by two pi, the measure of one full revolution
Why: The fraction of a revolution is the measure divided by one revolution, and one revolution is two pi.
Three eighths of a turn passes the quarter mark and stops midway through quadrant two, which is correct.
The three quarters that the wrong method produced is not meaningless — it is the fraction of a half revolution, and it is exactly the number you would get by converting to degrees and dividing by 180. Useful, but it is not what the question asked.
Faded example
Reduce twenty-three pi over four into the interval from zero to two pi.
Fill in the blanks
\frac216 - 7(2\pi) = \frac______ - \frac___\pi}___ = \frac___\pi}___
Why: Two pi written with denominator four is eight pi over four, so two revolutions is sixteen pi over four. Twenty-three minus sixteen leaves seven pi over four, which is seven eighths of a revolution and lands midway through quadrant four. Subtracting one more revolution would go negative, which confirms two was the right number to remove.
Estimation
An angle measures 10 radians.
Predict first
Roughly how many complete revolutions is that, and where does it end up?
Correct: About 1.6 revolutions, ending in quadrant three.
Why: One revolution is about 6.28, so 10 divided by 6.28 is about 1.59 revolutions. Removing one revolution leaves about 3.72 radians. Comparing against the landmarks, 3.72 is between pi at 3.14 and three pi over two at 4.71, which is quadrant three. The whole placement took one division and one comparison, with no conversion to degrees anywhere.
Explain it
A classmate converts every radian measure to degrees before doing anything, and it is costing them time on every problem.
Discussion prompt
Teach them the fraction-of-a-revolution shortcut in four sentences or fewer, and give them one example where it is obviously faster.
Hint: What cancels, and what is left?
Answer:
A usable answer: divide the measure by two pi and the pi always cancels, leaving a plain fraction. That fraction is the fraction of a full turn, so you can place the ray straight away. For example, thirteen pi over six divided by two pi is thirteen twelfths, which is one twelfth past a full turn, so it lands exactly where pi over six does — in quadrant one.
Converting instead would mean computing 13 times 180 divided by 6 to get 390 degrees, then subtracting 360 to get 30, then placing that. Three steps against one.
Comparison
Fill the blanks from memory. Every row is the same angle written two ways.
Comparison matrix
| Angle | Degrees | Radians |
|---|---|---|
| One full revolution | 360 | 2 pi |
| Half a revolution | 180 | pi |
| A right angle | 90 | pi over 2 |
| The angle whose arc equals the radius | about 57.3 | 1 |
| Coterminal step | add 360 k | add 2 pi k |
| Supplementary pair sums to | 180 | pi |
Only the numerals change. Every structural idea from Lesson 10.1a — standard position, quadrants, coterminal families, complementary and supplementary pairs — carries over unaltered.
Pattern
Whether the question says convert, graph, classify or find a coterminal angle, the same five moves cover it.
Step four is the habit worth building deliberately. Almost every student converts to degrees for the first two weeks and then stops, and the students who stop earlier find Chapter 10 substantially easier.
Check
A conversion. Do it on paper before you click.
Check your understanding
Convert 210 degrees to radians, exactly, in terms of pi.
Answer: A
Why: Multiplying 210 by pi over 180 gives 210 pi over 180. Both numbers divide by 30, leaving seven pi over six. A quick check on the size: 210 degrees is past 180 but well short of 270, and seven pi over six is just past pi and short of three pi over two, so both say quadrant three.
Check
Placing a terminal side. Use the fraction, not a conversion.
Check your understanding
In which quadrant does the terminal side of an angle measuring negative five pi over three lie?
Answer: A
Why: Five pi over three divided by two pi is five sixths, so this is five sixths of a revolution clockwise. Five sixths clockwise is one sixth short of a complete clockwise turn, which puts the terminal side just above the positive x-axis, in quadrant one. Equivalently, adding two pi gives pi over three, which is plainly quadrant one.
Check
The wrapping function. Compare against the landmarks.
Check your understanding
The real number t equals 8 is wrapped onto the unit circle starting from the point one comma zero. Where does the arc end?
Answer: B
Why: One revolution is about 6.283, so remove it: 8 minus 6.283 leaves about 1.717. Comparing against the landmarks, 1.717 sits between pi over two at 1.571 and pi at 3.142, so the endpoint is just past the positive y-axis, in quadrant two. Removing whole revolutions before comparing is the whole technique here.
Real world
A machinist is boring a circular slot in a plate of radius 150 millimetres. The drawing specifies the slot as running from 0 to 40 degrees. The CNC controller wants the arc length in millimetres.
Discussion prompt
Compute the arc length, and explain why a controller that accepted the angle in radians would need no conversion factor at all while one accepting degrees always does.
Hint: Use the definition of radian measure rearranged, and watch what happens to the constants.
Answer:
\[ 40^\circ \cdot \frac{\pi}{180^\circ} = \frac{2\pi}{9} \approx 0.6981 \]
\[ s = r\theta = 150 \cdot \frac{2\pi}{9} \approx 104.7 \text{ mm} \]
The formula s equals r theta is true only when theta is in radians, and that is not a convention — it is the definition of radian measure rearranged. In degrees the same computation needs s equals r times theta times pi over 180, and the factor pi over 180 is doing nothing but undoing the arbitrary choice of 360.
This is why machine controllers, physics engines and every calculus library work internally in radians and convert only at the human interface. The unit that makes the formula have no constant in it is the unit the machine should use.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
The same central angle is drawn in a circle of radius 1 and again in a circle of radius 100. How do the arc length and the radian measure compare between the two?
Correct: The arc length is 100 times larger; the radian measure is the same.
\[ \theta = \frac{s}{r} = \frac{100s_0}{100r_0} = \frac{s_0}{r_0} \]
Why: Radian measure is arc divided by radius. Scaling the circle by a factor of 100 multiplies the arc by 100 and the radius by 100, so the quotient is untouched. The arc itself, being a length, does scale. This is the whole reason the ratio is allowed to be called a measure of the angle: it reports something about the angle and refuses to report anything about the circle it was drawn in. It is also why the unit circle is worth singling out — there, and only there, the radian measure and the arc length are numerically equal.
Explain it
They are comfortable with degrees and think radians are a pointless complication invented to make exams harder.
Discussion prompt
In no more than five sentences and without using the word ratio, convince them that radians are the more natural unit. Give them one formula that is clean in radians and ugly in degrees.
Hint: Talk about measuring the arc with the radius as your ruler.
Answer:
A usable answer: instead of chopping the circle into 360 pieces because somebody once liked the number 360, measure the angle by laying the radius along the arc and counting how many fit. That count is the radian measure, and it comes from the circle rather than from a committee. Because it is a count of radius-lengths, the arc length is just the radius times the angle — s equals r theta, with nothing else in it.
In degrees that same formula is s equals r times theta times pi over 180, and the extra factor is there purely to undo the arbitrary 360. Every formula in calculus involving a trigonometric function has the same property, which is why nobody does calculus in degrees.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Conversion is fixed by writing the units into the fraction, so the wrong factor becomes visible instead of silent. Quadrant placement is fixed by dividing by two pi and reading the fraction of a revolution. Reduction is fixed by rewriting two pi with the angle's own denominator before subtracting, so it becomes a numerator subtraction. The invariance argument is fixed by writing the ratio for a scaled circle and watching the factor cancel. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
Draw a large circle in the middle of the page and mark its centre and one radius. Along the arc, lay off that radius length once and shade the central angle it cuts off, labelling it one radian, then write the definition of radian measure beside it. Around the outside of the circle, mark and label the eight angles zero, pi over four, pi over two, three pi over four, pi, five pi over four, three pi over two and seven pi over four, and beside each write the fraction of a revolution it represents. In the top left corner write both conversion factors with their units, and use each one once. In the bottom left corner, draw a short straight number line and a small circle beside it, and sketch the wrap, marking where the numbers zero, one, three and six land. In the bottom right corner, write the coterminal formula in radians and use it to reduce twenty-five pi over six into the interval from zero to two pi. Finally, circle the one angle on your page for which the radian measure and the arc length are the same number, and write why.
The circled item should be every angle on the page, provided the circle you drew has radius one — that is the point of the unit circle, and if you circled only one angle, reread the fourth section.
Recap
Five things, and the fourth one is the habit that makes the rest of Chapter 10 readable.
| If the question says | Your first move is |
|---|---|
| Convert to radians | Multiply by pi over 180, with the units written in |
| Convert to degrees | Multiply by 180 over pi, with the units written in |
| Which quadrant | Divide by two pi and read the fraction of a revolution |
| Find coterminal angles | Rewrite two pi with the angle's own denominator |
| Where does t land on the unit circle | Compare against 1.57, 3.14, 4.71 and 6.28 |
The next lesson cashes in the definition. Because radian measure is arc over radius, the arc length is radius times angle with no constant attached, and that one rearrangement gives arc length, sector area and the velocity of anything moving on a circle.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 700-706 — everything on these slides traces back here
Want this taught 1-on-1? Alexander tutors Trigonometry — $55/session, free consultation.