Angles as amounts of rotation rather than as static corners: degree measure as a fraction of a revolution, the degree-minute-second system and both conversions, complementary and supplementary pairs, oriented angles whose sign records direction, standard position and the quadrant an angle terminates in, and coterminal angles as the family of rotations sharing one terminal side.
Subject: Trigonometry · 65 slides · symbolic lesson
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Title
Trigonometry · Chapter 10 — Foundations of Trigonometry
§10.1 Angles and their Measure, pp. 693-700
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. 693-700 — the pages these objectives are drawn from
Warm-up
You have known what an angle is since primary school. This lesson replaces that definition with a better one, and it is worth seeing why the old one runs out.
Discussion prompt
A clock's minute hand moves from the 12 to the 3. Now let it keep going all the way round and stop at the 3 again. Both times it ended in the same place. Should those two motions count as the same angle? Argue either way.
Hint: There is no single right answer yet — the point is that the everyday word 'angle' does not decide it.
Answer:
The everyday picture of an angle as a corner cannot tell those two apart, because a corner has no memory of how it was made. That is fine in geometry, where you only ever care about the shape.
It is not fine here. Trigonometry describes things that turn — wheels, pendulums, alternating current — so the amount of turning has to survive into the answer. Stitz and Zeager therefore define the measure of an angle as the amount of rotation separating the rays, and by the end of this lesson you will have names for both readings: the two motions give different angles that happen to be coterminal.
Concept
Two rays sharing an initial point make an angle. But the picture alone does not say how much rotation separates them, and in trigonometry the rotation is the whole point. So the measure of an angle is defined as the amount of turning, measured as a fraction of one full revolution.
angle — The figure formed by two rays sharing a common initial point, called the vertex. Its measure is the amount of rotation that separates the two rays.
Once measure means rotation, three things follow immediately and they organise the rest of the lesson: rotation can exceed one full turn, rotation has a direction, and two different rotations can finish in the same place.
Figure (svg): A ray with its initial point marked P, then two rays sharing the initial point P to form an angle, and finally a straight angle where the two rays point in opposite directions
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 693-694
Section
Section 1
Concept
One complete revolution is defined to be 360 degrees, and parts of a revolution are measured proportionately. That is the entire definition, and every degree fact you know is it rearranged.
degree — One three-hundred-sixtieth of a complete revolution. An angle measuring a fraction f of a revolution measures f times 360 degrees.
Figure (svg): Three circles showing one full revolution as 360 degrees, a half revolution as 180 degrees, and a quarter revolution as 90 degrees with a small square marking the right angle
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 694-695
Picture it
The choice of 360 is usually credited to the Babylonians. It is convenient because 360 has a great many divisors, so common fractions of a turn come out as whole numbers.
Figure (svg): Three circles showing one full revolution as 360 degrees, a half revolution as 180 degrees, and a quarter revolution as 90 degrees with a small square marking the right angle
Nothing in trigonometry depends on the number 360 being special. In two lessons you will meet radians, which use a completely different constant and change none of the mathematics — only the labels.
Worked example
Do not reach for a formula. Read the definition and multiply.
\[ \text{Find the degree measure of an angle representing } \tfrac{2}{3} \text{ of a revolution, and one representing } \tfrac{1}{12}. \]
Write down what one revolution measures
Why: This is the only fact in play; everything else is arithmetic.
\[ 1\text{ revolution } = 360 ^\circ \]
Multiply the fraction by 360 for the first angle
Why: Measuring proportionately means exactly this multiplication.
\[ (\frac{2}{3}) (360) = 240 ^\circ \]
Multiply the fraction by 360 for the second angle
Why: Same move, different fraction.
\[ (\frac{1}{12}) (360) = 30 ^\circ \]
Classify each result
Why: 240 is more than 180, so it is a reflex amount of turning; 30 is between 0 and 90, so that angle is acute.
Figure (svg): The solution to Worked example a fraction of a turn as a degree measure shown as a ladder of expressions, one row per legal move
\[ \tfrac{2}{3}(360^\circ) = 240^\circ \qquad \tfrac{1}{12}(360^\circ) = 30^\circ \]
Verify: turn the answers back into fractions
Why: 240 divided by 360 is two thirds, and 30 divided by 360 is one twelfth. Both return the fractions we started from, so the multiplications went the right way.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 695-695
Sorting
Classification depends only on where the measure sits relative to 90 and 180.
Sort into buckets
Drop each angle measure into the right column.
Worked example
The same sentence, run backwards. This direction is the one that gets skipped, and it is the one radians will need.
\[ \text{What fraction of a revolution is } 30.5^\circ \text{? And } 0^\circ \text{?} \]
Divide the measure by 360
Why: If multiplying by 360 turns a fraction into degrees, dividing by 360 must undo it.
\[ \frac{30.5}{360} \]
Reduce the fraction
Why: Multiply top and bottom by 2 to clear the decimal, then cancel a factor of 2.
\[ \frac{61}{720} \]
Handle the second angle the same way
Why: Zero rotation is still a legitimate amount of rotation. The definition does not exclude it.
\[ \frac{0}{360} = 0 \]
Say what the second answer means
Why: An angle of zero degrees indicates no rotation at all — the two rays coincide.
\[ 0 ^\circ\text{ is no turning} \]
Figure (svg): The solution to Worked example an angle measure as a fraction of a turn shown as a ladder of expressions, one row per legal move
\[ 30.5^\circ = \tfrac{30.5}{360} = \tfrac{61}{720} \text{ of a revolution} \qquad 0^\circ = 0 \]
Verify: multiply back
Why: Sixty-one seven-hundred-twentieths of 360 is 61 times 360 divided by 720, which is 61 divided by 2, which is 30.5. It returns the original measure.
Trap
\[ 30.5^\circ = 30^\circ 50' \]
Read the decimal .5 as though the fractional part were out of 100
Why: Decimal notation trains you to read tenths and hundredths, so 0.5 gets converted as if it were 50 percent of something with 100 parts.
This treats a degree as if it had 100 minutes in it. It does not.
\[ 30.5^\circ = 30^\circ + 0.5^\circ \]
Convert the fractional degree using the real factor
Why: A degree holds 60 minutes, not 100, so the fractional part is multiplied by 60.
\[ 0.5^\circ \cdot \frac{60'}{1^\circ} = 30' \]
\[ 30.5^\circ = 30^\circ 30' \]
Half a degree is thirty minutes, exactly as half an hour is thirty minutes. The clock analogy is the one to trust here, not the decimal point.
Fill the middle
The angle representing five twelfths of a revolution.
Fill in the blanks
\tfrac5/12150\text___ = ___ \cdot 360^\circ = ___^\circ
Why: Measuring proportionately means multiplying the fraction of a revolution by 360 degrees. Five twelfths of 360 is 5 times 30, which is 150 degrees — an obtuse angle, which is the right size for a bit less than half a turn.
Prediction
An angle turns through seven eighths of a revolution.
Predict first
Without multiplying anything out, is the measure more or less than 300 degrees?
Correct: More than 300 degrees.
Why: Three hundred degrees is five sixths of a revolution, because 300 over 360 reduces to 5 over 6. Seven eighths is larger than five sixths, so the measure must exceed 300 degrees. Computing it confirms this: seven eighths of 360 is 315 degrees. Comparing fractions of a turn is faster than converting, and it is a useful habit for radians later.
Two truths and a lie
Rule out the false statement and say what makes it false.
Eliminate the wrong options
Rule out the statements that are TRUE. The survivor is the false one.
Survives elimination: B
Why: Statements A and C are both direct readings of the definition — zero rotation and one three-hundred-sixtieth of a rotation are both amounts of rotation. Statement B smuggles in a restriction the definition never made. Allowing measures beyond 360 degrees is the whole reason coterminal angles are worth a name, and it is what makes trigonometry able to describe something that keeps spinning.
Section
Section 2
Concept
There are two ways to subdivide a degree. The familiar one is decimal degrees. The other is the degree-minute-second system, in which one degree is divided into sixty minutes and each minute into sixty seconds.
DMS system — A measure written as degrees, minutes and seconds, where one degree equals sixty minutes and one minute equals sixty seconds, so that one degree equals three thousand six hundred seconds.
\[ 1^\circ = 60' \qquad 1' = 60'' \qquad \text{so} \quad 1^\circ = 3600'' \]
Minutes and seconds here are not units of time. The system is borrowed from the same Babylonian base-sixty counting that gave us the clock, which is why the arithmetic feels familiar.
Figure (svg): A bar showing one degree divided into sixty minutes, and one of those minutes divided again into sixty seconds, with the conversion factors labelled
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 695-696
Picture it
Every DMS conversion is one of two moves: multiply by sixty to go down a level, divide by sixty to come back up.
Figure (svg): A bar showing one degree divided into sixty minutes, and one of those minutes divided again into sixty seconds, with the conversion factors labelled
If you are going from a big unit to a small one the number should get bigger, so you multiply. That single check catches almost every DMS error.
Worked example
Example 10.1.1, part 1. Convert 111.371 degrees to the DMS system, rounded to the nearest second.
\[ \text{Convert } \alpha = 111.371^\circ \text{ to degrees, minutes and seconds.} \]
Split off the whole degrees
Why: The integer part is already in degrees and needs no conversion at all.
\[ 111 ^\circ + 0.371 ^\circ \]
Convert the leftover degrees to minutes
Why: Multiply by sixty minutes per degree, so the degree unit cancels.
\[ 0.371(60) = 22.26 \min \]
Split off the whole minutes
Why: Same move as before, one level down.
\[ 22 \min + 0.26 \min \]
Convert the leftover minutes to seconds
Why: Multiply by sixty seconds per minute.
\[ 0.26(60) = 15.6 \sec \]
Round to the nearest second and reassemble
Why: The question asked for the nearest second, so 15.6 becomes 16.
\[ 111 ^\circ 22 \min 16 \sec \]
Figure (svg): The solution to Worked example decimal degrees into DMS shown as a ladder of expressions, one row per legal move
\[ 111.371^\circ \approx 111^\circ 22' 16'' \]
Verify: convert the answer back
Why: Twenty-two minutes is 22 over 60 of a degree, or about 0.36667, and sixteen seconds is 16 over 3600, or about 0.00444. Their sum is about 0.37111, so the answer reads 111.371 degrees to three decimal places, matching the original.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 696-696
Matching
Each decimal degree measure on the left equals exactly one DMS measure on the right.
Match the pairs
Why: Half a degree is 30 minutes and a quarter degree is 15 minutes, exactly as with hours. For 1.1 degrees the leftover 0.1 degree times 60 is 6 minutes. For 2.005 degrees the leftover 0.005 degree times 3600 is 18 seconds, which is well under a minute — a good reminder that a seconds-only remainder is normal when the decimal part is small.
Worked example
Example 10.1.1, part 2. Convert 37 degrees 28 minutes 17 seconds to decimal degrees, to the nearest thousandth.
\[ \text{Convert } \beta = 37^\circ 28' 17'' \text{ to decimal degrees.} \]
Turn the minutes into a fraction of a degree
Why: Divide by sixty minutes per degree; now the minute unit cancels instead.
\[ \frac{28}{60} = 7 / 15 ^\circ \]
Turn the seconds into a fraction of a degree
Why: Divide by three thousand six hundred seconds per degree.
\[ 17 / 3600 ^\circ \]
Add all three pieces over a common denominator
Why: Thirty-six hundred works for all of them.
\[ \frac{133200 + 1680 + 17}{3600} \]
Do the division and round
Why: The exact value is a fraction; the question asked for a decimal.
\[ \frac{134897}{3600} = 37.4713... \]
Figure (svg): The solution to Worked example DMS back into decimal degrees shown as a ladder of expressions, one row per legal move
\[ 37^\circ 28' 17'' = \frac{134897}{3600}^\circ \approx 37.471^\circ \]
Verify: check the size against the pieces
Why: Twenty-eight minutes is a little under half a degree, and seventeen seconds is tiny, so the answer must be a little under 37.5. It is 37.471, which sits just below 37.5 exactly as expected.
Error analysis
A student converts 12 degrees 30 minutes 36 seconds to decimal degrees.
Annotate
On: \( 12^\circ + \frac{30}{60}^\circ + \frac{36}{60}^\circ = 12 + 0.5 + 0.6 = 13.1^\circ \)
The size check catches this instantly: 36 seconds is a hundredth of a degree, a genuinely tiny amount, so it cannot possibly add six tenths of a degree to the answer.
Estimation
You are asked to convert 44 degrees 59 minutes 59 seconds to decimal degrees.
Predict first
Before computing, what will the answer be closest to?
Correct: Just under 45.0 degrees.
Why: Fifty-nine minutes is one minute short of a full degree, and fifty-nine seconds is one second short of a full minute, so the measure is one second short of 45 degrees exactly. The decimal value is about 44.99997. Recognising a measure that is just short of a round number saves you from a conversion you did not need, and it is the same instinct that makes 59 minutes past the hour read as 'almost the next hour'.
Ranking
They are written in a mixture of notations on purpose.
Put in order
Why: Convert everything to decimals first: 17 degrees 6 minutes is 17.1, then 17.4, then 17 degrees 30 minutes is 17.5, then 17 degrees 45 minutes is 17.75, then 17.9. Mixed notation is the only thing making this hard, which is why the first move in any comparison is to put every measure into the same form.
Socratic
Decimal degrees are easier to compute with, and every calculator uses them internally.
Discussion prompt
Given that, why would anyone still use minutes and seconds? Name a field that does, and say what the base-sixty system buys them.
Hint: Think about who needs to state a position on the Earth very precisely, and out loud.
Answer:
Navigation and astronomy both still use DMS, and latitude and longitude are quoted that way on every marine chart. The advantage is that a whole number of seconds is already a very fine subdivision — one second of latitude is about thirty metres — so positions can be stated precisely without decimals, which are easy to mishear or mistranscribe over a radio.
Sixty also divides evenly by two, three, four, five and six, so common fractions of a degree come out as whole minutes. A third of a degree is exactly 20 minutes, whereas in decimal it is 0.3333 and never ends.
Section
Section 3
Concept
Two acute angles are complementary if their measures add to 90 degrees. Two angles are supplementary if their measures add to 180 degrees. Both names return constantly, and the second one returns in the Law of Sines at the very end of this course.
complementary angles — Two acute angles whose measures add to 90 degrees. Because both must be acute, an obtuse angle has no complement.
\[ \text{complementary: } \alpha + \beta = 90^\circ \qquad \text{supplementary: } \gamma + \theta = 180^\circ \]
The definition of supplementary allows a pair of right angles, or one acute and one obtuse angle — but never two acute angles, since their sum could not reach 180 degrees.
Figure (svg): Two diagrams side by side: complementary angles alpha and beta meeting to form a right angle, and supplementary angles gamma and theta meeting to form a straight line
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 696-697
Picture it
A complementary pair fills a right angle. A supplementary pair fills a straight line.
Figure (svg): Two diagrams side by side: complementary angles alpha and beta meeting to form a right angle, and supplementary angles gamma and theta meeting to form a straight line
That is why supplementary pairs turn up whenever a line is cut by another line, and why the ambiguous case of the Law of Sines will hinge on a supplementary pair sharing the same sine.
Worked example
Example 10.1.1, part 4. Find a supplementary angle for 111.371 degrees.
\[ \text{Find } \theta \text{ such that } 111.371^\circ + \theta = 180^\circ. \]
Write the defining equation
Why: Supplementary means the two measures add to 180 degrees. That sentence is the equation.
\[ 111.371 + \theta = 180 \]
Solve for the unknown angle
Why: Subtract the known measure from both sides.
\[ \theta = 180 - 111.371 \]
Do the subtraction
Why: Straight decimal arithmetic; nothing special is happening here.
\[ \theta = 68.629 ^\circ \]
Sanity-check the classification
Why: The original angle is obtuse, so its supplement must be acute, and 68.629 is indeed below 90.
\[ 68.629 ^\circ\text{ is acute} \]
Figure (svg): The solution to Worked example a supplement, in decimal degrees shown as a ladder of expressions, one row per legal move
\[ \theta = 180^\circ - 111.371^\circ = 68.629^\circ \]
Verify: add the pair
Why: 111.371 plus 68.629 is 180.000 exactly, which is the definition of supplementary, so the pair checks out.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 697-697
Discrimination
The wording rarely uses the words complementary or supplementary. It describes the picture instead.
Sort into buckets
Sort each situation by which pairing it is describing.
Worked example
Example 10.1.1, part 5. This is the one worth practising, because the borrowing is where it goes wrong.
\[ \text{Find } \gamma \text{ such that } 37^\circ 28' 17'' + \gamma = 90^\circ. \]
Set up the subtraction
Why: Complementary means the pair adds to 90 degrees, so the complement is 90 degrees minus the given angle.
\[ \gamma = 90 ^\circ - 37 ^\circ 28 \min 17 \sec \]
Rewrite 90 degrees so it has minutes and seconds to give away
Why: You cannot subtract 17 seconds from 0 seconds, so borrow one degree as 60 minutes.
\[ 90 ^\circ = 89 ^\circ 60 \min 0 \sec \]
Borrow again, one minute as sixty seconds
Why: Now there are seconds available in the seconds column.
\[ 90 ^\circ = 89 ^\circ 59 \min 60 \sec \]
Subtract column by column
Why: Degrees from degrees, minutes from minutes, seconds from seconds — no carrying is needed now.
\[ 52 ^\circ 31 \min 43 \sec \]
Figure (svg): The solution to Worked example a complement, in DMS, with borrowing shown as a ladder of expressions, one row per legal move
\[ \gamma = 89^\circ 59' 60'' - 37^\circ 28' 17'' = 52^\circ 31' 43'' \]
Verify: add the pair back
Why: Seventeen seconds plus forty-three seconds is sixty seconds, which carries as one minute. Twenty-eight plus thirty-one plus that carried minute is sixty minutes, which carries as one degree. Thirty-seven plus fifty-two plus that carried degree is ninety. The pair sums to exactly 90 degrees.
Trap
\[ \text{Find the complement of } 130^\circ. \]
Compute 90 degrees minus 130 degrees
Why: The subtraction is applied mechanically, without checking whether the definition even applies.
\[ 90^\circ - 130^\circ = -40^\circ \]
A negative measure is returned as though it answered the question. It does not — the definition of complementary requires both angles to be acute.
\[ \text{Find the complement of } 130^\circ. \]
Check the definition first
Why: Complementary angles are defined as a pair of ACUTE angles summing to 90 degrees.
An angle of 130 degrees is obtuse, so it is already larger than 90 degrees on its own. No acute angle can be added to it to reach 90.
The correct answer is that 130 degrees has no complement. It does have a supplement, namely 50 degrees, and that is almost certainly what the question meant to ask for.
Faded example
Find the complement of 61 degrees 14 minutes 52 seconds.
Fill in the blanks
90^\circ = 89^\circ 59' 60'' \quad\Longrightarrow\quad 89^\circ 59' 60'' - 61^\circ 14' 52'' = 28^@circ 45' 8''
Why: Rewriting 90 degrees as 89 degrees 59 minutes 60 seconds is the whole trick: it borrows one degree into the minutes column and one minute into the seconds column in a single step, so the subtraction goes column by column with no further carrying. Checking the answer, 52 plus 8 is 60 seconds, 14 plus 45 plus the carry is 60 minutes, and 61 plus 28 plus that carry is 90 degrees.
Prediction
An angle measures 74 degrees.
Predict first
Its complement and its supplement are both computed. Which of these is true of them?
Correct: The complement is acute and the supplement is obtuse.
Why: The complement is 90 minus 74, which is 16 degrees, and 16 is acute. The supplement is 180 minus 74, which is 106 degrees, and 106 is obtuse. The general rule behind this is worth keeping: a complement is always acute, because it is under 90 degrees by definition, while a supplement is obtuse exactly when the original angle is acute. So for any acute angle the pair always splits this way, one of each.
Counterexample
A student proposes: if two angles are supplementary, then exactly one of them is obtuse.
Discussion prompt
Find a pair of supplementary angles that makes this claim false, and then repair the claim so that it is true.
Hint: What is the one supplementary pair in which the two angles are equal?
Answer:
\[ 90^\circ + 90^\circ = 180^\circ \]
Two right angles are supplementary, and neither of them is obtuse. That single pair breaks the claim as stated.
The repair: if two angles are supplementary then at most one of them is obtuse, and if either is acute then the other must be obtuse. The equal-pair case is the only exception, and it is exactly the boundary case — which is why boundary cases are worth testing against any claim containing the word 'exactly'.
Section
Section 4
Concept
An oriented angle records not just how much rotation separates the rays but which way the rotation went. Counterclockwise rotation is recorded as a positive measure; clockwise rotation is recorded as a negative one.
standard position — An angle is in standard position when its vertex sits at the origin and its initial side lies along the positive x-axis. The remaining ray is called the terminal side.
Standard position is a convention, not a result. Its whole purpose is to make every angle comparable: once every angle starts in the same place, the only thing that distinguishes them is where the terminal side ends up.
Figure (svg): An angle in standard position on coordinate axes, with its vertex at the origin, its initial side along the positive x-axis, and its terminal side in the second quadrant, with the four quadrants labelled
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 697-698
Picture it
These two angles involve exactly the same amount of turning. Only the direction differs, and that alone puts the terminal side in a different quadrant.
Figure (svg): Two angles drawn in standard position: a positive angle of 120 degrees sweeping counterclockwise in green, and a negative angle of 120 degrees sweeping clockwise in red, both ending on different terminal sides
A negative measure never means a small angle or a backwards angle in any other sense. It means clockwise, full stop.
Worked example
Example 10.1.2. Graph each angle in standard position and say which quadrant its terminal side lies in.
\[ \alpha = 60^\circ, \qquad \beta = -225^\circ \]
Start both angles on the positive x-axis
Why: That is what standard position means; the starting ray is never in question.
Sweep 60 degrees counterclockwise
Why: The measure is positive, so the rotation runs counterclockwise. Sixty degrees is two thirds of the way to the vertical.
Sweep 225 degrees clockwise for the second angle
Why: The measure is negative, so the rotation runs the other way. Going clockwise, 180 degrees reaches the negative x-axis and another 45 continues past it.
Read off the quadrants
Why: Quadrants are numbered counterclockwise starting from the upper right, regardless of which way the angle was swept.
Figure (svg): Two angles in standard position: sixty degrees swept counterclockwise ending in the first quadrant, and negative two hundred twenty-five degrees swept clockwise ending in the second quadrant
\[ \alpha = 60^\circ \in \text{QI} \qquad \beta = -225^\circ \in \text{QII} \]
Verify: reach the same terminal side going the other way
Why: Sweeping counterclockwise instead, negative 225 degrees should land where positive 135 degrees does, since the two differ by one full revolution. And 135 degrees is between 90 and 180, which is quadrant II — the same answer.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 698-699
Sorting
Sweep each one in your head before answering. Watch the signs.
Sort into buckets
Sort each angle by where its terminal side lies.
Worked example
Example 10.1.2 continued. This is where defining measure as rotation earns its keep.
\[ \gamma = 450^\circ, \qquad \phi = -270^\circ \]
Subtract whole revolutions from 450 to see where it stops
Why: Each 360 degrees returns the ray to where it started, so only the remainder decides the terminal side.
\[ 450 - 360 = 90 \]
Place the terminal side for the first angle
Why: Ninety degrees puts the terminal side straight up the positive y-axis.
Sweep 270 degrees clockwise for the second angle
Why: Clockwise, 90 degrees reaches the negative y-axis, 180 reaches the negative x-axis, and 270 reaches the positive y-axis.
Name the classification
Why: Both terminal sides lie on an axis rather than inside a quadrant.
Figure (svg): Two quadrantal angles: four hundred fifty degrees swept counterclockwise past one full revolution, and negative two hundred seventy degrees swept clockwise, both ending on the positive y-axis
\[ 450^\circ \text{ and } -270^\circ \text{ are both quadrantal, terminating on the positive } y\text{-axis.} \]
Verify: check the difference between them
Why: The two measures differ by 450 minus negative 270, which is 720, and 720 is exactly two full revolutions. Angles differing by a whole number of revolutions must share a terminal side, so finding the same terminal side twice is exactly what should have happened.
Error analysis
A student is asked to draw negative 60 degrees in standard position.
Annotate
On: \( -60^\circ \;\longrightarrow\; \text{terminal side in QII} \)
The reliable habit is to draw the sweep arrow itself rather than just the final ray. If you have drawn the arrow going the right way, the quadrant takes care of itself.
Translation
Each phrase describes a rotation. Give the signed degree measure it corresponds to.
Match the pairs
Why: Multiply the number of turns by 360 and attach a minus sign for clockwise. A quarter of 360 is 90, made negative for clockwise. Three quarters is 270, left positive. One and a half turns is 540, which is more than one revolution and perfectly legal. Half a turn clockwise is negative 180, whose terminal side happens to match positive 180 — the two angles differ but their terminal sides do not.
Edge cases
The quadrants are open regions: an angle in quadrant one measures strictly between 0 and 90 degrees.
Discussion prompt
So what happens at exactly 90 degrees, and why does the course bother to give that case its own name instead of assigning it to a quadrant?
Hint: Think about what the terminal side does at that instant, and what its x-coordinate becomes.
Answer:
At exactly 90 degrees the terminal side lies along the y-axis, which is the boundary between quadrants one and two and belongs to neither. Assigning it to one of them would be arbitrary.
The name matters because of what happens next. In Lesson 10.3a the tangent of an angle is defined as a quotient with the x-coordinate of the terminal point downstairs, and on the y-axis that coordinate is zero. The quadrantal angles are precisely the angles where some of the six circular functions are undefined, so they need to be identifiable before those functions are even introduced.
Explain it to yourself
Two students are arguing. One says an angle of 400 degrees is 'really just' 40 degrees. The other says they are different angles.
Discussion prompt
Who is right, and in what sense is each of them right? Use the words terminal side and rotation in your answer.
Hint: Ask what question is being answered. There is more than one reasonable question here.
Answer:
They are both right, about different things. The terminal sides are identical, so any question whose answer depends only on where the ray ended — which quadrant is this in, what is its sine — gets the same answer for both.
But the rotations are different: 400 degrees is more than a full turn and 40 degrees is not. Any question about the turning itself — how far did the wheel travel, how much cable wound onto the drum — gets different answers.
Trigonometry needs both readings, which is why the next section gives the shared-terminal-side relationship its own name rather than pretending the two angles are equal.
Section
Section 5
Concept
Two angles in standard position are coterminal when they share a terminal side. Because one full revolution returns the ray to exactly where it started, adding or subtracting any whole number of revolutions produces a coterminal angle — and every coterminal angle arises that way.
coterminal angles — Two angles in standard position whose terminal sides coincide. Their measures differ by an integer multiple of 360 degrees.
\[ \theta \text{ is coterminal with } \alpha \iff \theta = \alpha + 360^\circ k, \quad k \in \mathbb{Z} \]
The integer k is not restricted in any way. It may be positive, negative, or zero, so every angle is coterminal with itself.
Figure (svg): Three angles sharing the same terminal side: 60 degrees, 420 degrees which is 60 plus one full revolution, and negative 300 degrees which is 60 minus one full revolution
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 699-700
Picture it
The four angles whose terminal sides lie on the axes are the ones the quadrant classification cannot reach.
Figure (svg): The four quadrantal angles zero, ninety, one hundred eighty and two hundred seventy degrees, each with its terminal side lying along an axis rather than inside a quadrant
Every quadrantal angle is coterminal with one of these four, which is why checking a quadrantal case usually means checking only four things rather than infinitely many.
Worked example
Example 10.1.2 asks for two coterminal angles for each angle — one positive, one negative.
\[ \text{Find one positive and one negative angle coterminal with } 117^\circ. \]
Write the general form
Why: Every coterminal angle is the original plus a whole number of revolutions, so one formula covers all of them.
\[ \theta = 117 + 360 k \]
Take k equal to one for a positive answer
Why: Adding one full revolution keeps the terminal side and increases the measure.
\[ 117 + 360 = 477 \]
Take k equal to negative one for a negative answer
Why: Subtracting one full revolution drops the measure below zero while keeping the terminal side.
\[ 117 - 360 = -243 \]
Check both against the quadrant
Why: The original is between 90 and 180 so it is in quadrant two, and both answers must be too.
Figure (svg): The solution to Worked example list coterminal angles shown as a ladder of expressions, one row per legal move
\[ 477^\circ = 117^\circ + 360^\circ \qquad -243^\circ = 117^\circ - 360^\circ \]
Verify: subtract the pairs
Why: 477 minus 117 is 360, and 117 minus negative 243 is also 360. Both differences are whole multiples of a revolution, which is exactly the condition for being coterminal.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 699-699
Elimination
Rule out three, and say what each one actually is.
Eliminate the wrong options
Which angle is coterminal with 40 degrees?
Survives elimination: C
Why: Negative 320 degrees differs from 40 degrees by exactly 360, so it is 40 degrees minus one full revolution: sweep clockwise 320 degrees and you arrive at the same ray you reach by sweeping counterclockwise 40 degrees. The single test for every one of these is whether the difference of the two measures is a whole multiple of 360, and only choice C passes it.
Worked example
This is the form the question almost always takes later in the course: not any coterminal angle, but the one lying in a stated window.
\[ \text{Find the angle coterminal with } 1125^\circ \text{ that lies in } [0^\circ, 360^\circ). \]
Decide how many revolutions to remove
Why: Divide the measure by 360 to see how many whole turns are hidden inside it.
\[ \frac{1125}{360} = 3.125 \]
Take the whole number of turns
Why: Three whole revolutions fit, so k must be negative three.
\[ k = -3 \]
Subtract that many revolutions
Why: Three revolutions is 1080 degrees.
\[ 1125 - 1080 = 45 \]
Confirm the result is in the window
Why: Forty-five degrees is at least 0 and strictly less than 360, so it is the one the question wanted.
\[ 45 ^\circ\text{ is in } [0, 360] \]
Figure (svg): The solution to Worked example the one coterminal angle in a given interval shown as a ladder of expressions, one row per legal move
\[ 1125^\circ - 3(360^\circ) = 1125^\circ - 1080^\circ = 45^\circ \]
Verify: rebuild the original
Why: Forty-five plus three full revolutions is 45 plus 1080, which is 1125 — the angle we started with. And since only whole revolutions were removed, the terminal side never moved.
Trap
\[ \text{An angle coterminal with } 65^\circ \text{ is } 65^\circ + 180^\circ = 245^\circ. \]
Add a fixed amount to reach a coterminal angle
Why: The idea of adding a constant is right, but the constant chosen is half a revolution rather than a whole one.
Sixty-five degrees is in quadrant one and 245 degrees is in quadrant three, so the terminal sides point in opposite directions. They are not the same ray.
\[ \text{An angle coterminal with } 65^\circ \text{ is } 65^\circ + 360^\circ = 425^\circ. \]
Add a whole revolution
Why: Only a complete revolution returns the terminal side to where it started; half a revolution leaves it pointing the opposite way.
\[ 425^\circ - 65^\circ = 360^\circ \quad \checkmark \]
Both angles land in quadrant one, on the same ray. The quadrant check is the fastest way to catch this error: coterminal angles are always in the same quadrant, so if the quadrant changed, something is wrong.
Fill the middle
Find the angle coterminal with negative 870 degrees that lies between 0 and 360 degrees.
Fill in the blanks
-870^\circ + 3(360^\circ) = -870^\circ + 1080^\circ = 210^\circ
Why: Dividing 870 by 360 gives about 2.4, so two revolutions are not quite enough to bring a negative measure up above zero and three are needed. Three revolutions is 1080 degrees, and negative 870 plus 1080 is 210 degrees, which sits in the required window and in quadrant three. A quick check: negative 870 is also in quadrant three, since it is 870 degrees clockwise, and coterminal angles never change quadrant.
Pattern
Here is the family of angles coterminal with 50 degrees, generated by stepping the integer k.
Step through it
The measures step by 360 each time. What stays exactly the same across all five, and what changes?
The terminal side and therefore the quadrant are identical in all five cases; only the amount of rotation differs. Every question about position gives one answer for the whole family, and every question about turning gives five different ones.
Missing information
A problem states: an angle in standard position has its terminal side in quadrant three. Find the angle.
Discussion prompt
Why can this not be answered as asked, and what is the smallest thing you could add to the statement to make the answer unique?
Hint: How many angles have a terminal side in quadrant three?
Answer:
Infinitely many angles have a terminal side in quadrant three — every measure strictly between 180 and 270 degrees does, and so does each of those plus or minus any whole number of revolutions. A quadrant pins down a region, not a ray.
Two additions would each make it unique. Either give the exact terminal side (say, the ray through the point with coordinates negative one and negative one) and restrict the measure to a single revolution such as the interval from 0 to 360 degrees; or give one specific measure and ask for a coterminal one in a stated window.
This is worth noticing now because from Lesson 10.2b onward, questions of the form 'find all angles satisfying this equation' always come with a window attached, and the window is doing exactly this job.
Comparison
Fill the blanks from memory before you scroll back. These six words are used without explanation for the rest of the course.
Comparison matrix
| Term | What it means | The test |
|---|---|---|
| Standard position | Vertex at the origin, initial side on the positive x-axis | Is the vertex at the origin and does the sweep start on the positive x-axis? |
| Terminal side | The ray the rotation ends on | Where did the arrow stop? |
| Positive measure | Counterclockwise rotation | Which way did the arrow sweep? |
| Quadrantal | Terminal side lies on an axis | Is the measure a multiple of 90 degrees? |
| Coterminal | Two angles sharing a terminal side | Is the difference of the measures a multiple of 360 degrees? |
| Supplementary | Two angles whose measures add to 180 degrees | Do they fill a straight line? |
Notice that four of the six tests are questions about a picture rather than about a formula. Drawing the angle is almost always faster than reasoning about the number.
Pattern
Whether the question says convert, classify, graph or find a coterminal angle, the same five moves cover it.
Step two is the one that makes the rest of the course tractable. Nearly every trigonometric question can be reduced to a question about an angle between 0 and 360 degrees, and doing that reduction first is almost always the right opening move.
Check
A DMS conversion. Do it on paper before you click.
Check your understanding
Convert 25.6 degrees to degrees and minutes.
Answer: B
Why: The whole part, 25 degrees, is already correct. The leftover 0.6 degrees is converted by multiplying by 60 minutes per degree, giving 36 minutes. The clock analogy confirms it: six tenths of an hour is 36 minutes, and a degree divides exactly as an hour does.
Check
Standard position and quadrants. Sweep it in your head first.
Check your understanding
In which quadrant does the terminal side of an angle measuring negative 145 degrees lie?
Answer: C
Why: A negative measure sweeps clockwise. Going clockwise from the positive x-axis, 90 degrees reaches the negative y-axis and 145 degrees continues 55 degrees past it, landing between the negative y-axis and the negative x-axis, which is quadrant three. Equivalently, adding a revolution gives 215 degrees, and 215 is between 180 and 270.
Check
Coterminal angles in a window.
Check your understanding
Which angle is coterminal with 1000 degrees and lies in the interval from 0 to 360 degrees?
Answer: B
Why: Dividing 1000 by 360 gives about 2.78, so two whole revolutions fit inside. Removing them gives 1000 minus 720, which is 280 degrees, and 280 lies in the required interval. Checking the difference, 1000 minus 280 is 720, exactly two revolutions.
Real world
A stepper motor turns a robot arm. Its controller accepts a target position as an angle in degrees and always drives the arm by the shortest route. The arm is currently at 20 degrees and is commanded to 700 degrees.
Discussion prompt
Where does the arm physically end up, how far does it actually rotate, and why would a controller that reduced every command into the interval from 0 to 360 degrees be dangerous for a machine that winds cable onto a spool?
Hint: Separate the two questions this lesson kept separating: where did it stop, and how much did it turn?
Answer:
\[ 700^\circ - 360^\circ = 340^\circ \]
The arm ends up physically indistinguishable from an arm at 340 degrees, because 700 and 340 are coterminal. Driven by the shortest route from 20 degrees, it sweeps 40 degrees clockwise rather than 680 degrees counterclockwise.
For a bare pointer that is fine. For a spool it is not: the commanded 680 degrees of counterclockwise rotation would have wound almost two full turns of cable on, and the controller instead unwound a tenth of a turn. The position is the same and the rotation is not, which is precisely the distinction the definition of an oriented angle was built to preserve.
Real controllers therefore track a cumulative angle that is deliberately never reduced, and reduce it only for display. That is the engineering version of keeping k in the coterminal formula rather than throwing it away.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Two angles in standard position have the same terminal side. Must they have the same measure?
Correct: No, they may differ by any whole number of revolutions — and yes if both are restricted to a single revolution.
\[ \theta_1 - \theta_2 = 360^\circ k, \quad k \in \mathbb{Z} \]
\[ 40^\circ, \; 400^\circ, \; -320^\circ \text{ all share one terminal side} \]
Why: Sharing a terminal side means being coterminal, and coterminal measures differ by an integer multiple of 360 degrees. So the measures need not be equal: 40 and 400 share a ray. But the difference is not arbitrary either, which rules out the third option. The fourth option is a true statement rather than the best answer: restricting to a single revolution does force the measures to agree, because two multiples of 360 apart cannot both fit in an interval of width 360. That is exactly why later lessons always state a window before asking you to find all angles.
Explain it
They have done geometry, so they know what a 40 degree angle looks like. They have never seen an angle of 400 degrees and think it is a typo.
Discussion prompt
In no more than five sentences, and using no symbols, convince them that an angle of 400 degrees is a real thing. Then tell them the one question they should ask to decide whether 400 and 40 can be treated as the same.
Hint: Give them a physical object that turns.
Answer:
A usable answer: in geometry an angle is a corner, and a corner cannot be bigger than a full turn. Here an angle measures how much something rotated, and a wheel can certainly rotate more than once. Four hundred degrees means one complete turn and then another forty degrees. It looks identical to forty degrees when you stop and photograph it, because the pointer ends in the same place.
The question to ask is: does this problem care where it stopped, or how far it turned? If it cares where it stopped, 400 and 40 are interchangeable. If it cares how far it turned, they are not, and the extra revolution is real.
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: DMS conversion is fixed by always asking whether the number should get bigger or smaller, which fixes the direction of every multiplication. Negative quadrants are fixed by drawing the sweep arrow instead of the final ray. Coterminal windows are fixed by dividing the measure by 360 first, so the number of revolutions to remove is read off rather than guessed at. Complementary versus supplementary is fixed by one picture each: a corner and a straight line. 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 set of coordinate axes filling the middle of the page and label the four quadrants. On it, draw and label these four angles in standard position, each with its sweep arrow showing direction: 60 degrees, 210 degrees, negative 45 degrees, and 450 degrees. Beside each one write which quadrant it terminates in, or the word quadrantal. In the top left corner, write the coterminal formula and use it to give one positive and one negative angle coterminal with 210 degrees. In the top right corner, write the two DMS conversion factors and convert 12.75 degrees to DMS. Along the bottom, draw a right angle split into two parts and a straight line split into two parts, and label each picture with the correct word. Finally, circle the one angle on your page whose terminal side lies on an axis, and write down what goes wrong if you try to name a quadrant for it.
The circled angle should be 450 degrees, which is coterminal with 90 degrees and lands on the positive y-axis. What goes wrong is that its terminal side is the boundary between two quadrants and belongs to neither — and, looking ahead, that its x-coordinate is zero, which will make the tangent undefined there in Lesson 10.3a.
Recap
Five things, and the fifth one is the one the rest of this course leans on hardest.
| If the question says | Your first move is |
|---|---|
| Convert to DMS | Split off the whole degrees, then multiply the remainder by 60 |
| Find the complement of an angle in DMS | Rewrite 90 degrees as 89 degrees 59 minutes 60 seconds |
| Graph the angle | Draw the sweep arrow, not just the terminal ray |
| Which quadrant | Reduce into the interval from 0 to 360 degrees first |
| Find a coterminal angle in an interval | Divide the measure by 360 to count the revolutions to remove |
The next lesson keeps every one of these ideas and changes only the unit. Radian measure replaces the arbitrary 360 with a number that comes out of the circle itself, and standard position, quadrants and coterminal angles all survive the change untouched.
Stitz & Zeager, College Trigonometry, Ch. 10 Foundations of Trigonometry — §10.1 Angles and their Measure §10.1, pp. 693-700 — everything on these slides traces back here
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