Sets up the chapter. Places angles in standard position with a sign convention, introduces radian measure as a ratio of arc length to radius, converts between degrees and radians, and derives the arc length and sector area formulas from the fraction-of-a-circle idea. Closes with coterminal angles, reference angles and angular speed.
Subject: Precalculus · 65 slides · symbolic lesson
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Title
Precalculus · Chapter 5 — Trigonometric Functions
§5.1 Angles, pp. 626-650
Objectives
Five things, each one you can check yourself on paper.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 626-650 — the pages these objectives are drawn from
Warm-up
Degrees work perfectly well for measuring angles. So why does every mathematics course past this point use radians instead?
Discussion prompt
A circle of radius 5 has an arc cut off by a 60 degree angle. Write a formula for the arc's length. Now imagine the angle were measured in some unit where the formula had no fractions in it.
Hint: What fraction of the whole circle is 60 degrees, and what is the whole circumference?
Answer:
Sixty degrees is a sixth of a turn, and the circumference is 2 pi times 5, so the arc is 10 pi over 6 — about 5.24. The formula needs the fraction 60 over 360 in it.
If instead the angle were measured so that a full turn was 2 pi units, that fraction would become the angle over 2 pi, and multiplying by 2 pi r would leave just r times the angle. No fractions and no constants.
That unit is the radian, and it was defined precisely so that arc length is radius times angle. Degrees are a historical convention; radians are the unit the geometry itself picks out.
Concept
One radian is the angle whose arc equals the radius. Because it is defined as a length divided by a length, it carries no units, and formulas that use it need no conversion constants.
radian — The measure of a central angle whose subtended arc has the same length as the radius. Equivalently, the arc length divided by the radius. A full turn is 2 pi radians.
\[ \theta=\frac{s}{r}, \qquad \text{so } s=r\theta \]
The absence of units is worth taking seriously rather than treating as a technicality. It is why radians can be multiplied by a length to give a length, and why every arc, area and speed formula in this section works only when the angle is in radians.
Figure (svg): A circle with an arc equal in length to its radius marked, and the angle subtending it labelled as one radian, showing that a radian is defined by a ratio of two lengths
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 626-632
Section
Section 1
Concept
An angle is in standard position when its vertex is at the origin and its initial side lies along the positive horizontal axis. Only the terminal side then varies.
Fixing the initial side is what makes angles comparable. Without the convention, an angle would be a shape rather than a number, and two angles of the same size drawn in different places would have no obvious relationship.
Figure (svg): An angle in standard position with its vertex at the origin and initial side along the positive horizontal axis, with the sweep arc and arrowhead showing the positive direction
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 626-633
Picture it
Same origin, same initial side, opposite directions.
Figure (svg): An angle in standard position with its vertex at the origin and initial side along the positive horizontal axis, with the sweep arc and arrowhead showing the positive direction
The arrowhead records the direction of the sweep. Both angles here have terminal sides in different places, and the sign is what distinguishes the two ways of getting to a given one.
Worked example
Locate the terminal side.
\[ \text{In which quadrant does } 200^\circ \text{ terminate? And } -45^\circ? \]
Recall the quadrant boundaries
Why: Ninety degree steps.
\[ 90, 180, 270, 360 \]
Place the first
Why: Two hundred is between 180 and 270.
Interpret the negative sign
Why: Sweep clockwise from the positive axis.
\[ \text{clockwise } 45 \]
Place the second
Why: Just below the positive horizontal axis.
Figure (svg): An angle in standard position with its vertex at the origin and initial side along the positive horizontal axis, with the sweep arc and arrowhead showing the positive direction
\[ 200^\circ: \text{ III}; \qquad -45^\circ: \text{ IV} \]
Verify: check the negative angle another way
Why: Adding 360 to negative 45 gives 315, which is between 270 and 360 — the fourth quadrant, agreeing. Converting a negative angle to a positive coterminal one is a reliable way to place it when the clockwise sweep is hard to picture.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 627-630
Sorting
Locate each terminal side.
Sort into buckets
Sort each angle.
Worked example
Terminal sides on an axis get their own name.
\[ \text{Where does } 270^\circ \text{ terminate?} \]
Sweep three quarters of a turn
Why: Counterclockwise from the positive axis.
Identify the position
Why: Pointing straight down.
Note it is on an axis
Why: Not inside any quadrant.
List the others
Why: Zero, 90, 180 and 270.
Figure (svg): The solution to Worked example a quadrantal angle shown as a ladder of expressions, one row per legal move
\[ \text{quadrantal, on the negative } y \text{ axis} \]
Verify: say why these matter later
Why: The quadrantal angles are where the trigonometric functions take their simplest values, and where some of them are undefined — §5.3 will find that the tangent has no value at 90 degrees. Recognising them now means those special cases arrive as expected rather than as surprises.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 630-633
Trap
\[ \text{an angle drawn from the positive } y \text{ axis, called } 30^\circ \]
Draw a 30 degree angle from a convenient reference line
Why: The angle's size is correct and it is drawn at the origin.
The initial side is taken to be the positive vertical axis rather than the horizontal one.
Standard position requires the initial side on the positive HORIZONTAL axis. Measured from there, the drawn angle is 60 degrees, not 30.
The convention is what makes the number meaningful. Without it, the same terminal side could be labelled with many different sizes depending on where the drawer chose to start.
Always start from the positive horizontal axis and sweep counterclockwise for a positive angle. That is the entire content of standard position, and every value in the rest of the chapter assumes it.
Prediction
An angle of 50 degrees is replaced by negative 50 degrees.
Predict first
What changes about its terminal side?
Correct: It reflects across the horizontal axis.
Why: Reversing the direction of the sweep from counterclockwise to clockwise, over the same size, places the terminal side the same distance the other side of the initial side — which is a reflection in the horizontal axis. This is why the sine changes sign and the cosine does not when an angle is negated, a fact §5.2 will use.
Faded example
An angle of 250 degrees is drawn in standard position.
Fill in the blanks
180 < 250 < 270, \textIIIleft, \text______
Why: Between 180 and 270 degrees the terminal side lies in the third quadrant, where both coordinates are negative — down and to the left. Knowing the quadrant will determine the signs of the trigonometric values in §5.2, which is why the classification is worth making automatic.
Socratic
Standard position is a convention rather than a mathematical necessity.
Discussion prompt
What would go wrong if angles could be drawn from any starting line?
Hint: Could two people describe the same terminal side with the same number?
Answer:
The same terminal side could be labelled with many different numbers, depending on where each person chose to start measuring. The number would describe a drawing rather than a position.
Fixing the initial side makes the angle's size and the terminal side's position equivalent information, so one determines the other. That is what lets a trigonometric function be defined as a function of the angle at all.
The choice of the positive horizontal axis is arbitrary — any fixed ray would do — but having some fixed ray is essential. It plays the same role as choosing an origin for the number line, which is equally arbitrary and equally necessary.
Section
Section 2
Concept
One radian is the angle whose arc equals its radius. Since it is a ratio of two lengths, it is a pure number and carries no units.
The dimensionlessness has a practical consequence worth noticing. Multiplying an angle in radians by a length gives a length; multiplying an angle in degrees by a length gives nothing meaningful. Every formula in this section is stated for radians because degrees would need a conversion factor buried inside it.
Figure (svg): A circle with an arc equal in length to its radius marked, and the angle subtending it labelled as one radian, showing that a radian is defined by a ratio of two lengths
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 633-638
Picture it
The thick arc has the same length as the radius, and the angle it subtends is one radian.
Figure (svg): A circle with an arc equal in length to its radius marked, and the angle subtending it labelled as one radian, showing that a radian is defined by a ratio of two lengths
About 57.3 degrees, which is an awkward number in degrees and exactly 1 in the unit it defines. That awkwardness is the price of a unit chosen by the geometry rather than by convention.
Worked example
Divide the arc by the radius.
\[ \text{A circle of radius } 4 \text{ has an arc of length } 10. \text{ Find the central angle in radians.} \]
Recall the definition
Why: Angle is arc over radius.
\[ \theta = \frac{s}{r} \]
Substitute
Why: Ten over four.
\[ \frac{10}{4} \]
Simplify
Why: Two and a half.
\[ 2.5\text{ radians} \]
Check the units cancel
Why: Length over length.
Figure (svg): A circle with an arc equal in length to its radius marked, and the angle subtending it labelled as one radian, showing that a radian is defined by a ratio of two lengths
\[ \theta=\frac{10}{4}=2.5 \text{ radians} \]
Verify: sanity-check the size
Why: A full turn is about 6.28 radians, so 2.5 radians is a little under half a turn — about 143 degrees. The arc of 10 is a bit more than twice the radius of 4, which matches an angle a bit more than 2 radians. Both readings agree.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 634-636
Faded example
A circle of radius 6 has an arc of length 15.
Fill in the blanks
\theta = \frac62.5} = ___ \text___
Why: Dividing the arc by the radius gives 2.5 radians. Note that this is the same angle as the earlier example with radius 4 and arc 10, because both have the same ratio — which is exactly what makes radian measure a property of the angle rather than of the circle.
Worked example
The circumference divided by the radius.
\[ \text{Show that a full turn is } 2\pi \text{ radians.} \]
Take the arc for a full turn
Why: The whole circumference.
\[ s = 2 \pi r \]
Apply the definition
Why: Arc over radius.
\[ \theta = 2 \pi r / r \]
Cancel
Why: The radius divides out.
\[ \theta = 2 \pi \]
Note what cancelled
Why: The result does not depend on r.
Figure (svg): The solution to Worked example why a full turn is 2 pi shown as a ladder of expressions, one row per legal move
\[ \theta=\frac{2\pi r}{r}=2\pi \]
Verify: notice why the radius had to cancel
Why: If the answer depended on the radius, radian measure would not be a property of the angle at all — a 90 degree angle would measure differently on different circles. The cancellation is what makes the definition well posed, and it happens because both the arc and the radius scale together.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 636-638
Error analysis
A student cancels units in an arc length calculation.
Annotate
On: \( s=r\theta=(4\text{ cm})(2\text{ rad})=8\text{ cm}\cdot\text{rad} \)
Radians are a ratio, so they disappear in any dimensional analysis. That is a feature: it is exactly why arc length equals radius times angle with nothing else in the formula.
Prediction
A radian is the angle whose arc equals the radius.
Predict first
Roughly how many degrees is that?
Correct: About 57 degrees.
Why: A full turn is 2 pi radians, which is about 6.28, and 360 divided by 6.28 is about 57.3. The awkwardness of that number in degrees is the point: the radian is defined by the geometry rather than chosen to be convenient, and it is degrees that turn out to be the arbitrary unit.
Sorting
Formulas derived from the arc-over-radius definition do.
Sort into buckets
Sort each formula.
Explain it to yourself
Radians carry no units, unlike metres or seconds.
Discussion prompt
Explain why, and why it matters for the arc length formula.
Hint: What two quantities is a radian the ratio of?
Answer:
A radian is an arc length divided by a radius — a length over a length — so the units cancel and what remains is a pure number.
That is what lets the formula multiply an angle by a length and get a length. Multiplying metres by a genuine unit would produce something with mixed dimensions, which is not an arc length.
Degrees would need a conversion factor buried in the formula, since a degree is not a ratio of anything. The radian was defined to make these formulas clean, and every one of them in this section works only in radians for exactly that reason.
Section
Section 3
Concept
A half turn is 180 degrees and also pi radians. Every conversion follows from that single equality by multiplying by the fraction that cancels the unit you have.
\[ 180^\circ=\pi \text{ radians} \]
The unit-cancelling rule removes any need to remember which fraction goes which way. Writing the conversion as a fraction with the starting unit underneath makes the correct choice mechanical, exactly as in any other unit conversion.
Figure (svg): The degree-radian conversion shown from the fact that a full turn is both 360 degrees and 2 pi radians, with the two conversion fractions derived from it
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 638-642
Picture it
The two fractions are reciprocals, and cancelling decides which to use.
Figure (svg): The degree-radian conversion shown from the fact that a full turn is both 360 degrees and 2 pi radians, with the two conversion fractions derived from it
Neither has to be memorised separately. Write the half-turn equality, form the fraction that cancels your starting unit, and the direction takes care of itself.
Worked example
Multiply by the fraction that cancels degrees.
\[ \text{Convert } 135^\circ \text{ to radians.} \]
Choose the fraction
Why: Degrees must cancel, so they go underneath.
\[ \times \pi / 180 \]
Multiply
Why: One hundred thirty five over one eighty.
\[ 135 \pi / 180 \]
Simplify the fraction
Why: Both divide by 45.
\[ 3 \pi / 4 \]
Check the size
Why: A bit under a full pi.
Figure (svg): The degree-radian conversion shown from the fact that a full turn is both 360 degrees and 2 pi radians, with the two conversion fractions derived from it
\[ 135^\circ=\tfrac{3\pi}{4} \text{ radians} \]
Verify: check against the half turn
Why: Half a turn is pi radians and 135 degrees is three quarters of 180, so the answer should be three quarters of pi — which it is. Comparing against the half turn is the quickest sanity check on any degree-to-radian conversion.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 639-640
Faded example
Convert 240 degrees.
Fill in the blanks
240 \cdot \frac44 = \frac___\pi}___, \text___ ___\pi/3
Why: Two hundred forty over 180 simplifies to four thirds, so the answer is four pi over 3. Checking: that is four thirds of a half turn, and 240 degrees is indeed four thirds of 180. The comparison with the half turn is the fastest check available.
Worked example
The other fraction, chosen the same way.
\[ \text{Convert } \tfrac{5\pi}{6} \text{ radians to degrees.} \]
Choose the fraction
Why: Radians must cancel, so they go underneath.
\[ \times 180 / \pi \]
Multiply
Why: The pi symbols cancel.
\[ 5 \times 180 / 6 \]
Simplify
Why: One eighty over six is thirty.
\[ 5 \times 30 \]
Compute
Why: One hundred fifty.
\[ 150 ^\circ \]
Figure (svg): The solution to Worked example radians to degrees shown as a ladder of expressions, one row per legal move
\[ \tfrac{5\pi}{6}=150^\circ \]
Verify: check against the half turn again
Why: Five sixths of a half turn is five sixths of 180, which is 150 — agreeing. Note the pi cancelled entirely, which always happens when converting an exact multiple of pi to degrees, so a leftover pi in the answer signals an error.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 640-642
Trap
\[ 60^\circ \times \frac{180}{\pi} = \frac{10800}{\pi} \text{ radians} \]
Multiply by a conversion fraction
Why: One of the two standard fractions is applied.
The result is an enormous number described as an angle in radians.
The degrees did not cancel. Starting in degrees, the fraction must have degrees underneath, so it is pi over 180.
That gives 60 pi over 180, which simplifies to pi over 3, about 1.05 radians. A full turn is only about 6.28 radians, so an answer in the thousands is visibly wrong.
Write the fraction so the starting unit cancels. That removes the choice entirely, and a quick size check against 6.28 catches any that slip through.
Matching
All are fractions of a half turn.
Match the pairs
Why: Each is a simple fraction of 180 degrees, so its radian measure is the same fraction of pi. These four recur constantly through the rest of the chapter and are worth knowing without computation, since almost every exact trigonometric value uses one of them.
Estimation
An angle measures 2 radians.
Predict first
Roughly how many degrees is that?
Correct: A bit under 120 degrees.
Why: A half turn is about 3.14 radians, so 2 radians is a bit under two thirds of a half turn, which is a bit under 120 degrees. The exact value is about 114.6. Estimating against the half turn catches an inverted conversion immediately, since that would have given about 3.5 degrees.
Prediction
An exact multiple of pi radians is converted to degrees.
Predict first
What happens to the pi?
Correct: It cancels, leaving a number with no pi.
Why: The conversion fraction has pi underneath, so a pi in the starting value cancels against it. Any leftover pi in a degree answer signals that the wrong fraction was used. Conversely, a degree-to-radian answer usually has a pi in it, which is the same check run backwards.
Section
Section 4
Concept
A central angle cuts off a fraction of the circle equal to the angle over a full turn. Multiplying that fraction by the circumference gives the arc, and by the area gives the sector's area.
\[ s=r\theta, \qquad A=\tfrac{1}{2}r^2\theta \]
Deriving them is genuinely quicker than recalling which has the half and which does not. Take the fraction, multiply by the whole, and cancel — the half in the area formula appears because the circle's area has a pi and the circumference has a 2 pi.
Figure (svg): A circular sector with its arc length and area formulas shown, both requiring the angle in radians, alongside the fraction-of-the-circle reasoning that produces them
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 642-646
Picture it
Both come from the fraction-of-the-circle idea shown beneath.
Figure (svg): A circular sector with its arc length and area formulas shown, both requiring the angle in radians, alongside the fraction-of-the-circle reasoning that produces them
The warning is the operative part. Both formulas assume radians, and using degrees in them gives an answer wrong by a factor of about 57.
Worked example
Convert to radians first if necessary.
\[ \text{Find the arc cut off by } 60^\circ \text{ on a circle of radius } 9. \]
Convert the angle to radians
Why: Sixty is a third of 180.
\[ \frac{\pi}{3} \]
Apply the formula
Why: Radius times angle.
\[ 9 \times \pi / 3 \]
Simplify
Why: Nine over three is three.
\[ 3 \pi \]
Approximate if wanted
Why: Three times 3.14159.
\[ \text{about } 9.42 \]
Figure (svg): A circular sector with its arc length and area formulas shown, both requiring the angle in radians, alongside the fraction-of-the-circle reasoning that produces them
\[ s=9\cdot\tfrac{\pi}{3}=3\pi\approx 9.42 \]
Verify: check against the circumference
Why: The full circumference is 18 pi, and 60 degrees is a sixth of a turn, so the arc should be 18 pi over 6, which is 3 pi — agreeing. Checking against the fraction of the whole is the derivation itself, run as a check.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 643-644
Faded example
Find the arc for an angle of pi over 6 on a circle of radius 12.
Fill in the blanks
s = 12 \cdot \frac26.28 = ___\pi \approx ___
Why: Twelve over six is 2, so the arc is 2 pi, about 6.28. Checking against the circumference of 24 pi: pi over 6 is a twelfth of a turn, and 24 pi over 12 is 2 pi — agreeing. The fraction check is the derivation and is worth using as a verification.
Worked example
The same fraction applied to the area instead.
\[ \text{Find the area of a sector with angle } \tfrac{\pi}{4} \text{ and radius } 6. \]
Apply the formula
Why: Half r squared theta.
\[ (\frac{1}{2}) (36) (\frac{\pi}{4}) \]
Simplify the radius part
Why: Half of 36.
\[ 18 \times \pi / 4 \]
Simplify
Why: Eighteen over four.
\[ 9 \pi / 2 \]
Approximate
Why: About 14.14.
\[ \text{about } 14.1 \]
Figure (svg): The solution to Worked example sector area shown as a ladder of expressions, one row per legal move
\[ A=\tfrac{1}{2}(36)\tfrac{\pi}{4}=\tfrac{9\pi}{2}\approx 14.1 \]
Verify: check against the whole circle
Why: The full area is 36 pi, and pi over 4 radians is an eighth of a turn, so the sector should be 36 pi over 8, which is 4.5 pi — the same as nine pi over two. The fraction check confirms both the formula and the arithmetic.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 645-646
Error analysis
A student computes an arc length without converting.
Annotate
On: \( s=r\theta=(9)(60)=540 \)
Check the arc against the circumference. An arc longer than the whole circle is impossible, and that comparison catches a missing conversion immediately.
Prediction
A sector's angle is held fixed and its radius is doubled.
Predict first
What happens to the arc length and the area?
Correct: The arc doubles and the area quadruples.
Why: The arc formula is linear in the radius and the area formula involves the radius squared, so doubling the radius doubles one and quadruples the other. This is §3.9's scaling with exponents 1 and 2, applied to a geometric setting.
Sorting
Arc length or sector area.
Sort into buckets
Sort each question.
Explain it to yourself
The area formula has a half and the arc formula does not.
Discussion prompt
Derive both and explain the difference.
Hint: What are the circumference and the area of a full circle?
Answer:
Both are the fraction theta over 2 pi, times the whole. For the arc the whole is the circumference, 2 pi r, and the 2 pi cancels leaving r theta.
For the area the whole is pi r squared, and dividing by 2 pi leaves r squared over 2 — hence the half. The half is what survives from the mismatch between the 2 pi in the fraction and the single pi in the area.
So the difference is not arbitrary and does not need memorising. Deriving takes two lines and is more reliable than recalling which formula carries the half, which is the standard thing people get wrong.
Section
Section 5
Concept
Angles differing by whole turns share a terminal side. The reference angle is the acute angle to the horizontal axis, and it is what makes the special values transferable between quadrants.
The reference angle is the tool that makes §5.2's table of special values usable everywhere. All the exact values are computed for acute angles, and the reference angle plus the quadrant's sign pattern extends them to the whole circle.
Figure (svg): Three angles sharing a terminal side, differing by full turns, illustrating that coterminal angles look identical in standard position
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 646-650
Picture it
All three angles end in the same place.
Figure (svg): Three angles sharing a terminal side, differing by full turns, illustrating that coterminal angles look identical in standard position
Adding or subtracting a full turn changes the number and not the position, which is why every trigonometric value repeats with period 2 pi — the fact §6.1 will build on.
Worked example
Add or subtract turns until it lands in the wanted range.
\[ \text{Find an angle between } 0 \text{ and } 2\pi \text{ coterminal with } \tfrac{19\pi}{4}. \]
Express a full turn with the same denominator
Why: Two pi is eight pi over four.
\[ 2 \pi = 8 \pi / 4 \]
Subtract one turn
Why: Nineteen minus eight.
\[ 11 \pi / 4 \]
Still too large, subtract another
Why: Eleven minus eight.
\[ 3 \pi / 4 \]
Check it is in range
Why: Between zero and 2 pi.
Figure (svg): Three angles sharing a terminal side, differing by full turns, illustrating that coterminal angles look identical in standard position
\[ \tfrac{3\pi}{4} \]
Verify: check the difference is whole turns
Why: Nineteen pi over four minus three pi over four is sixteen pi over four, which is four pi — exactly two full turns. So the two angles are genuinely coterminal, and any trigonometric function takes the same value at both.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 647-648
Faded example
Find an angle between 0 and 360 degrees coterminal with negative 100 degrees.
Fill in the blanks
-100 + 360 = 260 \text___
Why: Adding one full turn brings the angle into the standard range, giving 260 degrees, which terminates in the third quadrant. Negative 100 sweeps clockwise into the third quadrant too, confirming they share a terminal side.
Worked example
The two are related by the radius, in radians.
\[ \text{A wheel of radius } 0.3 \text{ m spins at } 4 \text{ radians per second. Find the rim's speed.} \]
Identify the relationship
Why: Linear is radius times angular.
\[ v = r \omega \]
Substitute
Why: Point three times four.
\[ 0.3 \times 4 \]
Compute
Why: One point two.
\[ 1.2 \]
Attach units
Why: Metres per second.
\[ 1.2 m / s \]
Figure (svg): The solution to Worked example linear and angular speed shown as a ladder of expressions, one row per legal move
\[ v=r\omega=(0.3)(4)=1.2 \text{ m/s} \]
Verify: check the units work
Why: Metres times radians per second gives metres per second, because radians are dimensionless. Had the angular speed been in degrees per second the formula would need a conversion factor, which is exactly the point of using radians — the units come out right with nothing extra.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 648-650
Trap
\[ v=r\omega=(0.3)(229)=68.7 \text{ m/s} \]
Convert the angular speed to degrees per second and substitute
Why: Four radians per second is about 229 degrees per second, which is used directly.
The rim speed is reported as 68.7 metres per second.
The formula requires radians per second. Using degrees inflates the answer by the number of degrees in a radian, about 57.
The correct speed is 1.2 metres per second. A sanity check catches it: a wheel 60 centimetres across spinning less than once a second cannot have a rim moving at 68 metres per second.
Every formula in this section requires radians, for the same reason: they multiply an angle by a length, which only works when the angle is dimensionless.
Prediction
An angle of 210 degrees terminates in the third quadrant.
Predict first
What is its reference angle?
Correct: 30 degrees.
Why: The reference angle is the acute angle to the horizontal axis, and 210 is 30 past 180. That 30 degrees is what makes the exact values transferable: the trigonometric values at 210 have the same sizes as at 30, with signs set by the third quadrant.
Sorting
Coterminal means differing by whole turns.
Sort into buckets
Sort each angle.
Real world
Anything that rotates has both a linear and an angular speed.
Discussion prompt
Two points on a spinning record, one near the centre and one at the rim. How do their speeds compare?
Hint: Which speed do they share and which do they not?
Answer:
They share the angular speed: both complete a full turn in the same time, since they are on the same rigid disc.
Their linear speeds differ, because linear speed is radius times angular speed. The point at the rim has a larger radius and therefore moves faster along its circle.
This is why the outer edge of a record wears faster, why the tip of a helicopter blade approaches the speed of sound while the hub barely moves, and why a runner in an outer lane starts further forward. One rotation, many speeds, and the radius is what distinguishes them.
Comparison
Fill the blanks from memory. One is a convention and one is picked out by the geometry.
Comparison matrix
| degrees | radians | |
|---|---|---|
| a full turn | 360 | 2 pi |
| defined by | an ancient convention | arc length over radius |
| carries units | yes, a degree symbol | no, it is a pure ratio |
| arc length formula | needs a conversion factor | s = r theta, with nothing extra |
| used in calculus | never | always |
The fourth row is why the fifth is true. Calculus formulas for trigonometric functions are clean in radians and carry stray constants in degrees, so the choice is made for you from Chapter 12 onward.
Pattern
Five steps, and the second prevents most of the errors.
Step 5's check is the cheapest error-catcher in the section. An arc longer than the circumference, or a sector larger than the circle, means a missing conversion.
Check
Cancel the starting unit.
Check your understanding
Convert 210 degrees to radians.
Answer: A
Why: Multiplying by pi over 180 gives 210 pi over 180, which simplifies to seven pi over six. Checking: 210 is seven sixths of 180, so the answer should be seven sixths of pi.
Check
The formula needs radians.
Check your understanding
A circle has radius 8 and a central angle of pi over 4. What is the arc length?
Answer: A
Why: Multiplying the radius by the angle gives 8 times pi over 4, which is 2 pi. Checking against the circumference of 16 pi: pi over 4 is an eighth of a turn, and 16 pi over 8 is 2 pi.
Check
Whole turns only.
Check your understanding
Which angle is coterminal with 100 degrees?
Answer: A
Why: Subtracting a full turn from 100 gives negative 260, so the two differ by exactly 360 degrees and share a terminal side. Coterminal angles must differ by a whole number of turns.
Real world
Radians are not a classroom convenience; they are what physics and computing use.
Discussion prompt
Why does every programming language's sine function expect radians rather than degrees?
Hint: What formulas are used to compute a sine internally?
Answer:
Sine is computed internally from a power series whose coefficients are clean only in radians. In degrees every term would carry a factor of pi over 180 raised to a power.
More fundamentally, the calculus of trigonometric functions is clean in radians: the derivative of sine is cosine exactly, with no constant, and only because radians are used. In degrees a factor of pi over 180 appears in every derivative.
So radians are not a preference. They are the unit in which the mathematics has no debris, and everything downstream — physics formulas, signal processing, graphics libraries — inherits that choice. Passing degrees to a sine function is one of the most common bugs in beginner code for exactly this reason.
Commit first
State your confidence along with your answer.
Predict first
Why do the arc length and sector area formulas require radians?
Correct: Because radians are dimensionless, so an angle can multiply a length.
Why: A radian is an arc length divided by a radius, so it is a pure number and multiplying it by a length gives a length. A degree is not a ratio of anything, so using it requires a conversion factor inside the formula. The clean formulas are the reason the unit was defined this way.
Explain it
The test of understanding is being able to say why, not just what.
Discussion prompt
Explain to a classmate what a radian is and why it is worth the trouble of a second angle unit.
Hint: Start from the arc, not from the conversion.
Answer:
A radian is the angle whose arc equals the radius. Wrap a length equal to the radius around the circle's edge, and the angle it subtends is one radian — about 57 degrees, which sounds arbitrary and is not.
It is worth the trouble because it makes arc length equal radius times angle, with no constant. Every formula relating an angle to a length becomes clean, because a radian is a ratio and carries no units.
The strongest argument comes later: in calculus the derivative of sine is cosine only in radians, and every trigonometric formula in physics assumes them. A good explanation says that degrees are the arbitrary unit here, which is the reverse of most people's intuition.
Exit ticket
One honest answer, so the next lesson can start in the right place.
Predict first
Which idea from this lesson would you most want to see again?
Correct: Any of these is a legitimate answer; the useful one is the honest one.
Why: There is no correct choice here. The second is the conceptual centre and makes the fourth make sense rather than needing memorisation. The first is what §5.2 will assume without comment, since every value there is read off a terminal side in standard position.
Connect it up
One page, drawn from memory, is worth more than rereading the section.
Draw it
Draw a circle with a marked radius and an arc of the same length, labelling the angle as one radian. Beside it, write the half-turn equality and both conversion fractions, noting which cancels which unit. Then draw a sector and derive both the arc length and area formulas from the fraction-of-a-circle idea, showing where the half comes from.
If your derivation shows the half arising from the mismatch between 2 pi and pi, you will not need to remember which formula carries it.
Recap
Five things, and the second is what the rest of the chapter is built on.
| if you remember one thing | it should be this |
|---|---|
| about radians | a ratio of two lengths, so no units and no constants in the formulas |
| about converting | cancel the unit you have; the direction follows |
| about the formulas | derive from the fraction of a circle rather than memorising |
| about checking | compare against the whole circumference or area |
Section 5.2 puts a point on the unit circle and defines the cosine and sine as its coordinates, which is where the angles of this section finally acquire functions attached to them.
OpenStax, Precalculus, §5.1 Angles §5.1, pp. 626-650 — everything on these slides traces back here
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