Angles in standard position with any real measure, coterminal angles, the definition of a radian, conversion between degrees and radians, the degree and radian measures of the special angles, and the arc length and area of a sector.
Subject: Algebra 2 · 65 slides · symbolic lesson
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Title
Algebra 2 · Chapter 13 — Trigonometric Ratios and Functions
Define General Angles and Use Radian Measure
Objectives
Five outcomes. Angles of any size, in either of two units.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 859-863 — the lesson these objectives are drawn from
Warm-up
In Lesson 13.1 every angle was an acute angle inside a right triangle.
Discussion prompt
What would a 240 degree angle mean? There is no triangle with such an angle. And what could a NEGATIVE angle possibly measure?
Hint: Think about rotation rather than about a corner.
Answer:
Stop thinking of an angle as a corner and start thinking of it as a rotation. Fix one ray, spin another away from it, and the angle measures how far you spun.
Two hundred forty degrees is two thirds of a full turn. Negative just means you spun the other way, and nothing stops you from turning more than once.
\[ 240^\circ, \; 500^\circ, \; -50^\circ \; \text{ all make sense} \]
Concept
In standard position an angle has its vertex at the origin and its initial side on the positive x-axis. The measure records how far the terminal side rotated: counterclockwise is positive, clockwise negative, and more than one revolution is allowed. Angles can be measured in degrees or in radians.
radian — The measure of an angle in standard position whose terminal side intercepts an arc equal in length to the radius. Because a circle's circumference is 2 pi r, a full circle is 2 pi radians, so 180 degrees equals pi radians.
\[ 180^\circ = \pi \text{ radians} \]
The unit radians is often omitted, so a measure written as five pi over four means five pi over four radians.
Figure (svg): Two columns comparing degree measure with radian measure
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 859-861
Section
Section 1
Concept
An angle is in standard position when its vertex sits at the origin and its initial side lies along the positive x-axis. The terminal side's rotation gives the measure, positive counterclockwise and negative clockwise.
\[ 240^\circ = 180^\circ+60^\circ; \quad 500^\circ = 360^\circ+140^\circ \]
The terminal side may make more than one complete rotation, which is what lets the measure be any real number.
Figure (svg): Three angles drawn in standard position
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 859-859 — Angles in Standard Position
Picture it
Example 1: 240, 500 and negative 50 degrees.
Figure (svg): Three angles drawn in standard position
The first passes the negative x-axis, the second completes a full turn and keeps going, and the third rotates the wrong way on purpose.
Worked example
Example 1, all three parts.
\[ \text{Draw } 240^\circ, \; 500^\circ, \text{ and } -50^\circ \text{ in standard position.} \]
First: locate 240 degrees
Why: Two forty is 60 more than 180.
\[ 60\text{ past the negative x-axis} \]
Second: reduce 500 degrees
Why: Five hundred is 140 more than 360.
\[ \text{one full turn plus } 140 \]
Third: read the sign
Why: Negative means clockwise.
\[ 50\text{ below the positive x-axis} \]
Draw each terminal side
Why: Initial side always on the positive x-axis.
Figure (svg): Three angles drawn in standard position
\[ 240^\circ, \; 500^\circ, \; -50^\circ \]
Verify: name the quadrant of each
Why: Two forty lies in quadrant three, 500 lands in quadrant two, and negative 50 in quadrant four. Subtracting or adding 360 until the measure lies between 0 and 360 is the fastest way to find the quadrant of any angle.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 859-859
Fill the middle
Example 1b.
Fill in the blanks
500^\circ = 360^\circ + 140^\circ
Why: One hundred forty, so the terminal side makes one complete turn and then 140 degrees more, landing in quadrant two.
Worked example
Guided Practice 1 to 4.
\[ \text{Draw } 65^\circ, \; 230^\circ, \; 300^\circ \text{ and } 740^\circ \text{ in standard position.} \]
First: 65 degrees
Why: Under 90.
Second: 230 degrees
Why: Fifty past 180.
Third: 300 degrees
Why: Sixty short of 360.
Fourth: 740 degrees
Why: Seven forty minus 720 is 20.
\[ \text{two full turns plus } 20 \]
Figure (svg): The solution to Worked example four more angles shown as a ladder of expressions, one row per algebraic move
\[ 65^\circ, \; 230^\circ, \; 300^\circ, \; 740^\circ \]
Verify: check the last one's reduction
Why: Seven hundred forty minus two times 360 is 740 minus 720, which is 20 — so the terminal side sits just above the positive x-axis. Two whole revolutions leave no trace in the picture, which is precisely the observation the next idea builds on.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 860-860
Trap
\[ -50^\circ \]
Rotate 50 degrees counterclockwise and label it negative
Why: The sign is treated as a label rather than a direction.
\[ \text{terminal side in quadrant one} \quad \text{(wrong)} \]
The sign IS the direction. A negative measure means the terminal side rotated clockwise, putting it in quadrant four here.
\[ -50^\circ \text{: rotate clockwise} \]
Let the sign choose the direction of rotation
Why: Positive counterclockwise, negative clockwise.
\[ \text{terminal side in quadrant four} \]
Adding 360 confirms it: negative 50 plus 360 is 310, which is indeed in quadrant four.
Sorting
Reduce to between 0 and 360 first.
Sort into buckets
Sort each angle by the quadrant of its terminal side.
Seven hundred forty reduces to 20, so it shares quadrant one with 65 despite looking nothing like it. Reduction first, quadrant second, always.
Matching
Rotation, direction and turns.
Match the pairs
Why: Each description translates into arithmetic on 360: add it for each full turn, and let the sign carry the direction. The picture never distinguishes 20 from 740, which is what makes coterminal angles a genuine idea rather than a technicality.
Prediction
Commit before reasoning.
Predict first
The terminal sides of 20 and 740 degrees are identical. Why bother distinguishing the two measures at all?
Correct: Because the measure records the ROTATION performed, not just the final direction, and the number of turns often matters physically.
\[ 740^\circ = 2(360^\circ)+20^\circ \]
Why: A wheel that has turned 740 degrees has done something different from one that has turned 20, even though a photograph of the two is identical. Allowing measures beyond one revolution is what lets an angle describe a process over time rather than only a static direction, which is the whole reason Chapter 14 can use these angles to model waves and rotating machinery.
Section
Section 2
Concept
Two angles are coterminal when their terminal sides coincide. An angle coterminal with a given one is found by adding or subtracting any multiple of 360 degrees.
\[ \theta \pm 360^\circ k, \quad k \text{ an integer} \]
There are infinitely many coterminal angles for any given one, which is why questions ask for one positive and one negative rather than for the answer.
Figure (svg): Three different measures sharing one terminal side
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 860-860 — Coterminal angles
Picture it
Example 2a: angles coterminal with negative 45 degrees.
Figure (svg): Three different measures sharing one terminal side
Adding 360 gives 315, subtracting it gives negative 405, and adding 720 gives 675 — all pointing the same way.
Worked example
Example 2, both parts.
\[ \text{Find one positive and one negative angle coterminal with } -45^\circ \text{ and with } 395^\circ. \]
First: add a full turn
Why: Negative 45 plus 360.
\[ 315 ^\circ \]
First: subtract a full turn
Why: Negative 45 minus 360.
\[ -405 ^\circ \]
Second: subtract a full turn
Why: Three ninety-five minus 360.
\[ 35 ^\circ \]
Second: subtract two full turns
Why: Three ninety-five minus 720.
\[ -325 ^\circ \]
Figure (svg): Three different measures sharing one terminal side
\[ 315^\circ, -405^\circ; \qquad 35^\circ, -325^\circ \]
Verify: check each difference is a multiple of 360
Why: Three fifteen minus negative 45 is 360; negative 405 minus negative 45 is negative 360; 35 minus 395 is negative 360. Any pair of coterminal angles differs by a whole number of full turns, which is both the definition and the check.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 860-860
Fill the middle
Example 2a.
Fill in the blanks
-45^\circ+360^\circ = 315^\circ
Why: Three hundred fifteen degrees, in quadrant four — the same direction as negative 45, reached the long way round.
Worked example
Guided Practice 1 to 4.
\[ \text{Find a positive and a negative coterminal angle for } 65^\circ, 230^\circ, 300^\circ \text{ and } 740^\circ. \]
First: 65 degrees
Why: Add and subtract 360.
\[ 425\text{ and } -295 \]
Second: 230 degrees
Why: Add and subtract 360.
\[ 590\text{ and } -130 \]
Third: 300 degrees
Why: Add and subtract 360.
\[ 660\text{ and } -60 \]
Fourth: 740 degrees
Why: Subtract 720; subtract 1080.
\[ 20\text{ and } -340 \]
Figure (svg): The solution to Worked example four more shown as a ladder of expressions, one row per algebraic move
\[ 425^\circ/-295^\circ; \; 590^\circ/-130^\circ; \; 660^\circ/-60^\circ; \; 20^\circ/-340^\circ \]
Verify: note that many answers are correct
Why: For 65 degrees, 785 and negative 655 are equally valid, and so are infinitely many others. That is why the instruction says find ONE positive and ONE negative, and why any answer differing from these by a multiple of 360 is also right.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 860-860
Error analysis
A student looks for an angle coterminal with 65 degrees.
Annotate
On: \( 65^\circ+180^\circ = 245^\circ \)
Adding 180 produces the ray pointing the opposite way, which is a useful angle for other purposes but is not coterminal with the original.
Fill the middle
Guided Practice 4.
Fill in the blanks
740^\circ-2(360^\circ) = 740^\circ-720^\circ = 20^\circ
Why: Twenty degrees. One subtraction of 360 would leave 380, still above a full turn, so two were needed.
Sorting
Differ by a multiple of 360.
Sort into buckets
Sort each angle.
Two twenty-five points exactly opposite 45 and 135 points at right angles to it. Both are related to 45, but neither shares its terminal side.
Prediction
Commit before reasoning.
Predict first
How many angles are coterminal with 65 degrees?
Correct: Infinitely many, one for every integer number of full turns added.
\[ 65^\circ+360^\circ k, \quad k = \dots,-2,-1,0,1,2,\dots \]
Why: The list runs 65, 425, 785, 1145 upward and negative 295, negative 655, negative 1015 downward, forever in both directions. Every one of them describes the same terminal side. That is why a question can only ask for one of each sign, and why any answer differing from a printed one by a multiple of 360 deserves full credit.
Section
Section 3
Concept
One radian is the angle whose terminal side cuts off an arc equal in length to the radius. Since the circumference is 2 pi times the radius, a full circle contains exactly 2 pi radians.
\[ 360^\circ = 2\pi \text{ radians} \;\Longrightarrow\; 180^\circ = \pi \text{ radians} \]
One radian is about 57.3 degrees, an awkward number in degree terms, which is exactly the point: the unit is defined by the circle, not by human convention.
Figure (svg): One radian defined by an arc equal in length to the radius
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 860-860 — Radian measure
Picture it
The definition, drawn.
Figure (svg): One radian defined by an arc equal in length to the radius
Laying that arc around the circle takes 2 pi copies, so 2 pi radians make a full turn and pi radians make a half turn.
Worked example
The definition on page 860.
\[ \text{Show that a full circle measures } 2\pi \text{ radians.} \]
Write the circumference
Why: Two pi times the radius.
\[ C = 2 \pi r \]
Recall the definition
Why: One radian corresponds to an arc of length r.
\[ \text{arc } r\text{ means } 1\text{ radian} \]
Divide
Why: How many arcs of length r fit around.
\[ 2 \pi r\text{ divided by } r \]
Conclude
Why: The r cancels.
\[ 2 \pi\text{ radians} \]
Figure (svg): One radian defined by an arc equal in length to the radius
\[ \frac{2\pi r}{r} = 2\pi \]
Verify: check the size of one radian
Why: Two pi is about 6.28, so one radian is 360 over 6.28, about 57.3 degrees. A right angle of 90 degrees is therefore about 1.57 radians, which is pi over 2 — and pi over 2 really is about 1.571.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 860-860
Fill the middle
The definition.
Fill in the blanks
\frac2___ = \frac______ = ___\pi \text___
Why: Two pi radians. The radius cancels, which is why the answer does not depend on the size of the circle.
Worked example
The conversion rule on page 860.
\[ \text{From } 360^\circ = 2\pi, \text{ derive the conversion factors.} \]
Halve both sides
Why: One eighty equals pi.
\[ 180 ^\circ = \pi\text{ radians} \]
Divide to isolate the ratio
Why: Pi radians over 180 degrees.
Invert it
Why: One eighty degrees over pi radians.
Note both equal 1
Why: The two sides are the same quantity.
Figure (svg): The solution to Worked example the half-turn identity shown as a ladder of expressions, one row per algebraic move
\[ \times\frac{\pi}{180^\circ}; \qquad \times\frac{180^\circ}{\pi} \]
Verify: check which factor to use by inspection
Why: Converting 180 degrees should give pi, and 180 times pi over 180 does give pi. Using the other factor would give 180 times 180 over pi, about 10,313 — absurd. Testing a factor on a value you already know is faster than memorising which is which.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 860-860
Trap
\[ \pi \text{ radians} \]
Read pi as the number 3.14 and stop
Why: The symbol is treated as an ordinary quantity with no unit.
\[ \pi \approx 3.14^\circ \quad \text{(wrong)} \]
Pi radians is a half turn, 180 degrees. The number 3.14 counts RADIANS, and each radian is about 57.3 degrees.
\[ \pi \text{ radians} = 180^\circ \]
Keep track of which unit the number counts
Why: Radians and degrees are different sizes.
\[ 3.14 \text{ radians} \approx 3.14(57.3^\circ) \approx 180^\circ \]
The two calculations agree, which is the point. Confusing the units is the single commonest error in this lesson.
Comparison
Fill the blanks. The same four rotations.
Comparison matrix
| Rotation | Degrees | Radians |
|---|---|---|
| Full turn | 360 | 2 pi |
| Half turn | 180 | pi |
| Quarter turn | 90 | pi/2 |
| One radian | about 57.3 | 1 |
Degrees are chosen for convenience, radians by the circle itself. The last row shows why radians look strange in degree terms: nothing about 57.3 is memorable, and nothing about it needs to be.
Sorting
A right angle is pi over 2, about 1.57 radians.
Sort into buckets
Sort each radian measure.
One radian is under a right angle and two radians is over it, so the boundary sits between them at about 1.571. Keeping that decimal in mind makes radian measures easy to picture.
Prediction
Commit before reasoning.
Predict first
Degrees are convenient whole numbers. Why does mathematics prefer radians?
Correct: Because arc length is simply r times theta in radians, with no conversion constant anywhere.
\[ s = r\theta \text{ (radians)}; \quad s = \frac{\pi r\theta}{180} \text{ (degrees)} \]
Why: In degrees the arc length would be r theta times pi over 180, and the sector area would carry the same clutter. Radians are defined precisely so that constant becomes 1. The same simplification runs much deeper than this chapter: in calculus, the derivative of sine is cosine only when the angle is in radians, and picks up a stray factor otherwise.
Section
Section 4
Concept
Since 180 degrees equals pi radians, the ratio of the two is 1. Multiply a degree measure by pi over 180 to get radians, and a radian measure by 180 over pi to get degrees.
\[ \times\frac{\pi}{180^\circ}; \qquad \times\frac{180^\circ}{\pi} \]
The special angles in the first quadrant are worth memorising in both units, since everything else is a multiple of them.
Figure (svg): The first-quadrant special angles in both degree and radian measure
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 860-861 — Converting Between Degrees and Radians
Picture it
The first quadrant, and the multiples that follow from it.
Figure (svg): The first-quadrant special angles in both degree and radian measure
Five values carry the whole diagram: 0, pi over 6, pi over 4, pi over 3 and pi over 2. Every other special angle repeats one of those denominators.
Worked example
Example 3, both parts.
\[ \text{Convert } 125^\circ \text{ to radians and } -\tfrac{\pi}{12} \text{ radians to degrees.} \]
First: choose the factor
Why: Degrees to radians uses pi over 180.
\[ 125 \times \pi / 180 \]
First: reduce
Why: One twenty-five over 180 divides by 5.
\[ 25 \pi\text{ over } 36 \]
Second: choose the factor
Why: Radians to degrees uses 180 over pi.
\[ -\pi / 12 \times 180 / \pi \]
Second: simplify
Why: The pi cancels; 180 over 12 is 15.
\[ -15 ^\circ \]
Figure (svg): The first-quadrant special angles in both degree and radian measure
\[ \frac{25\pi}{36}; \qquad -15^\circ \]
Verify: check both against a landmark
Why: Twenty-five pi over 36 is a little under 25 over 36 of a half turn, which is a little under 125 over 180 of 180 degrees — consistent. And negative 15 degrees is a small clockwise angle, matching negative pi over 12, which is one twelfth of a half turn.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 861-861
Fill the middle
Example 3a.
Fill in the blanks
125^\circ\!\left(\frac25___\right) = \frac______ = \frac___\pi}___
Why: Twenty-five pi over 36. Both 125 and 180 divide by 5, and nothing further is common.
Worked example
Guided Practice 5 to 8.
\[ \text{Convert } 135^\circ, \; -50^\circ, \; \tfrac{5\pi}{4}, \; \tfrac{\pi}{10}. \]
First: 135 degrees
Why: One thirty-five over 180 reduces by 45.
\[ 3 \pi\text{ over } 4 \]
Second: negative 50 degrees
Why: Fifty over 180 reduces by 10.
\[ -5 \pi\text{ over } 18 \]
Third: five pi over four
Why: Times 180 over pi.
\[ 225 ^\circ \]
Fourth: pi over ten
Why: One eighty over 10.
\[ 18 ^\circ \]
Figure (svg): The solution to Worked example four conversions shown as a ladder of expressions, one row per algebraic move
\[ \tfrac{3\pi}{4}; \; -\tfrac{5\pi}{18}; \; 225^\circ; \; 18^\circ \]
Verify: check the first and third are consistent
Why: One thirty-five degrees is 3 pi over 4, and 5 pi over 4 is 225 degrees — the two differ by exactly 90 degrees, or pi over 2, in both units. Conversions that preserve the differences between angles are almost certainly right.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 861-861
Trap
\[ 125^\circ \times \frac{180^\circ}{\pi} \]
Grab whichever factor comes to mind
Why: The two factors look interchangeable.
\[ = \frac{22500}{\pi} \approx 7162 \quad \text{(absurd)} \]
An angle of 125 degrees cannot become 7162 of anything. The units must cancel: degrees on top must meet degrees underneath.
\[ 125^\circ \times \frac{\pi}{180^\circ} = \frac{25\pi}{36} \]
Put the unit you are LEAVING underneath
Why: So that it cancels.
\[ \approx 2.18 \text{ radians} \]
Two point one eight radians is a bit over a right angle's 1.57, which fits a 125 degree angle exactly.
Fill the middle
Guided Practice 7.
Fill in the blanks
\frac225___\!\left(\frac______\right) = 5\!\left(\frac______\right) = ___^\circ
Why: Two hundred twenty-five degrees, in quadrant three. The pi cancels, leaving straightforward arithmetic.
Matching
The special angles.
Match the pairs
Why: The pattern is easy to hold: the denominator counts how many of that angle fit into 180 degrees. Six thirties, four forty-fives, three sixties — and 135 is three of the forty-fives, hence 3 pi over 4.
Prediction
Commit before reasoning.
Predict first
In the radian measure pi over 6, what does the 6 tell you?
Correct: That six of these angles fit into a half turn, so the angle is 180 over 6, or 30 degrees.
\[ \frac{\pi}{6} = \frac{180^\circ}{6} = 30^\circ \]
Why: Since pi radians is a half turn, pi over 6 is a sixth of a half turn, which is 30 degrees. The same reading works throughout: pi over 4 is a quarter of 180, or 45; pi over 3 is a third of 180, or 60. Converting a simple radian measure needs no formula at all once you read the denominator this way, and it explains why 3 pi over 4 is three of those 45 degree pieces.
Section
Section 5
Concept
A sector is the region bounded by two radii and an arc. When the central angle is measured in radians, the arc length is r theta and the area is one half r squared theta.
\[ s = r\theta; \qquad A = \tfrac{1}{2}r^2\theta \]
Both formulas fail if the angle is in degrees. Convert first, always.
Figure (svg): A quarter-circle sector with its arc length and area
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 861-862 — Arc Length and Area of a Sector
Picture it
Example 4: a sector of radius 180 feet with a 90 degree central angle.
Figure (svg): A quarter-circle sector with its arc length and area
Converting 90 degrees to pi over 2 first is what makes both formulas usable; feeding in 90 would give answers wrong by a factor of about 57.
Worked example
Example 4.
\[ \text{A sector has } r = 180 \text{ ft and central angle } 90^\circ. \text{ Find the arc length and area.} \]
Convert the angle
Why: Ninety times pi over 180.
\[ \frac{\pi}{2}\text{ radians} \]
Find the arc length
Why: One eighty times pi over 2.
\[ 90 \pi,\text{ about } 283\text{ feet} \]
Find the area
Why: Half of 180 squared, times pi over 2.
\[ 8100 \pi \]
Approximate
Why: Eight thousand one hundred times 3.1416.
\[ \text{about } 25, 400\text{ square feet} \]
Figure (svg): A quarter-circle sector with its arc length and area
\[ s = 90\pi \approx 283; \quad A = 8100\pi \approx 25{,}400 \]
Verify: check against the whole circle
Why: A full circle of radius 180 has circumference 360 pi and area 32,400 pi. The sector is a quarter turn, and 90 pi is a quarter of 360 pi while 8100 pi is a quarter of 32,400 pi. Both check out.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 862-862
Fill the middle
Example 4.
Fill in the blanks
s = r\theta = 180\!\left(\frac90___\right) = ___\pi
Why: Ninety pi feet, about 283. That is exactly a quarter of the full circumference 360 pi, as a quarter turn should be.
Worked example
Guided Practice 9.
\[ \text{Repeat with the fence } 220 \text{ feet from home plate.} \]
Keep the angle
Why: Still a quarter turn.
\[ \frac{\pi}{2}\text{ radians} \]
Find the arc length
Why: Two twenty times pi over 2.
\[ 110 \pi,\text{ about } 346\text{ feet} \]
Find the area
Why: Half of 220 squared, times pi over 2.
\[ 12, 100 \pi \]
Approximate
Why: Twelve thousand one hundred times 3.1416.
\[ \text{about } 38, 000\text{ square feet} \]
Figure (svg): The solution to Worked example a larger field shown as a ladder of expressions, one row per algebraic move
\[ s = 110\pi \approx 346; \quad A = 12100\pi \approx 38{,}000 \]
Verify: compare the two fields
Why: The radius grew by a factor of 220 over 180, about 1.22, and the fence grew by the same factor: 346 over 283 is 1.22. But the area grew by 1.22 squared, about 1.49, and 38,000 over 25,400 is indeed about 1.50. Length scales with r; area scales with r squared.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 862-862
Error analysis
A student computes the arc length of a 90 degree sector of radius 180.
Annotate
On: \( s = r\theta = 180(90) = 16{,}200 \text{ feet} \)
The answer is too large by a factor of 180 over pi, about 57.3 — the number of degrees in a radian. Comparing the arc with the full circumference catches this instantly.
Fill the middle
Example 4.
Fill in the blanks
A = \tfrac8100___r^2\theta = \tfrac______(32400)\!\left(\frac______\right) = ___\pi
Why: Eight thousand one hundred pi square feet, about 25,400. Halving twice divides 32,400 by 4, matching the quarter-turn angle.
Ranking
Smallest first.
Put in order
Why: The values are about 283, 346, 1131 feet, then 25,400 and 38,000 square feet. Lengths and areas are not comparable quantities, but within each kind the ordering follows the radius — linearly for length, quadratically for area.
Prediction
Commit before reasoning.
Predict first
Why is a sector's area one half r squared theta?
Correct: Because the sector is the fraction theta over 2 pi of the whole circle, and that fraction times pi r squared gives it.
\[ \frac{\theta}{2\pi}\,\pi r^2 = \tfrac{1}{2}r^2\theta \]
Why: The sector occupies theta out of the circle's 2 pi radians, so its area is theta over 2 pi times pi r squared, and the pi cancels to leave one half r squared theta. The arc length formula comes the same way from the circumference. Both formulas are just proportions, and both look clean only because the angle is measured in radians.
Comparison
Fill the blanks. Two units, one set of angles.
Comparison matrix
| Question | Degrees | Radians |
|---|---|---|
| Full circle | 360 | 2 pi |
| Convert to the other by | multiplying by pi/180 | multiplying by 180/pi |
| Right angle | 90 | pi/2 |
| Works in s = r theta | no | yes |
Degrees survive because they are easy to picture, and radians because they make the geometry formulas clean. The conversion is a single multiplication either way.
Pattern
Position, reduce, convert.
Both sector formulas are wrong by a factor of about 57 if the angle is left in degrees.
Check
Add or subtract a full turn.
Check your understanding
Which angle is coterminal with -45 degrees?
Answer: A
Why: -45 + 360 = 315, and 315 - (-45) = 360, a full revolution.
Check
Put the unit you are leaving underneath.
Check your understanding
What is 125 degrees in radians?
Answer: A
Why: 125 times pi/180 is 125 pi/180, which reduces by 5 to 25 pi/36.
Check
Convert the angle first.
Check your understanding
A sector has radius 180 feet and central angle 90 degrees. What is its arc length?
Answer: A
Why: 90 degrees is pi/2 radians, so s = 180(pi/2) = 90 pi, about 283 feet.
Real world
A bicycle wheel has a radius of 13 inches. It turns through 5 complete revolutions plus 40 degrees more.
Discussion prompt
How far does the bicycle travel, and what single angle in radians describes the rotation?
Hint: Distance travelled is the arc length rolled out along the ground.
Answer:
\[ \theta = 5(360^\circ)+40^\circ = 1840^\circ \]
\[ 1840^\circ\!\left(\frac{\pi}{180}\right) = \frac{92\pi}{9} \approx 32.1 \text{ radians} \]
\[ s = r\theta = 13\!\left(\frac{92\pi}{9}\right) \approx 417 \text{ inches} \approx 34.8 \text{ feet} \]
The bicycle travels about 34.8 feet.
This is where allowing angles past 360 pays off. Reducing 1840 degrees to its coterminal 40 would describe the wheel's final ORIENTATION correctly and its DISTANCE not at all — the five lost revolutions are 5 times 2 pi times 13, about 408 of those 417 inches. Rotation that accumulates is exactly what the general angle was invented for, and it is why a wheel's odometer counts turns rather than positions.
Commit first
Answer, then rate your confidence honestly.
Predict first
Is an angle of 2 radians larger or smaller than a right angle?
Correct: Larger — a right angle is pi over 2, about 1.57 radians, so 2 radians is past it.
\[ 2 \text{ rad} \approx 114.6^\circ > 90^\circ \]
Why: The temptation is to compare 2 with 90, but those numbers count different units. One radian is about 57.3 degrees, so 2 radians is about 114.6 degrees, comfortably past the right angle and into quadrant two. Keeping the landmark pi over 2 equals about 1.57 in mind converts this from a calculation into a glance, and it is the single most useful number to memorise about radians.
Explain it
They know degrees and have never heard of a radian.
Discussion prompt
In four sentences or fewer, explain what a radian is.
Hint: Talk about wrapping the radius around the edge.
Answer:
Take a circle and lay its own radius along the edge as an arc. The angle at the centre that this arc cuts out is one radian.
Since the edge is 2 pi radii long, a full circle is 2 pi radians, or about 6.28. It is a unit the circle chooses for itself, rather than one we picked.
Exit ticket
Name the weakest spot before you close the deck.
Predict first
Which of these would you least want handed to you cold?
Correct: Whichever you picked is tonight's ten minutes, and each has a one-line fix.
Why: For drawing, reduce to between 0 and 360 first and let the sign choose the direction. For coterminal angles, add or subtract 360 as many times as needed. For the conversion factor, put the unit you are leaving underneath so it cancels. For sector formulas, make the conversion the first written line of every solution.
Connect it up
Paper. Fifteen minutes.
Draw it
Build a general-angles page. Top left: draw four angles in standard position, one under 90, one over 180, one negative and one past 360, labelling the quadrant of each. Top right: pick one angle and list four measures coterminal with it, two positive and two negative, showing the multiple of 360 used for each. Middle: draw a circle, mark the arc of length r and label the angle one radian, then write the chain from circumference to 2 pi radians to 180 equals pi. Bottom left: draw the first-quadrant special angles with both measures, and beside them convert two degree measures and two radian measures of your own choosing. Bottom right: draw a sector, convert its angle to radians, and compute both its arc length and its area, checking each against the whole circle.
If any sector answer was larger than the whole circle it came from, the angle was left in degrees. Convert and redo it.
Recap
Five things, and angles that are no longer trapped in triangles.
| If you see | Then |
|---|---|
| A measure over 360 or negative | Reduce by whole turns to find the quadrant |
| A request for a coterminal angle | Add or subtract a multiple of 360 |
| A conversion to make | Put the unit you are leaving underneath |
| pi over n radians | It is 180 over n degrees |
| A sector problem | Convert to radians before either formula |
| An arc longer than its circle | The angle was left in degrees |
Lesson 13.3 puts these general angles to work: a point on the terminal side defines the six ratios for any angle at all, not merely the acute ones.
McDougal Littell Algebra 2 (Texas Edition), Ch. 13 Trigonometric Ratios and Functions — Lesson 13.2 Define General Angles and Use Radian Measure §13.2, pp. 859-863 — everything on these slides traces back here
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