Sections 3.1 – 3.7
Angles, Rotation, and the Radian
What Trigonometry Studies · Angles as Rotations · Degree Review · Radian Measure · Conversion · Coterminal · Reference Angles
Section 3.1
What Trigonometry Studies
Trigonometry is the branch of mathematics that studies relationships between angles and distances. The word comes from the Greek words for triangle and measure. In its oldest form, trigonometry was indeed about triangles: it allowed surveyors, navigators, and astronomers to find unknown distances and angles by measuring what was accessible and computing what was not.
But trigonometry as studied in precalculus and calculus is far larger than triangle measurement. Once angles are extended from triangles to rotations, and once trigonometric ratios are extended from triangles to coordinates on a circle, trigonometry becomes a system for describing anything that repeats, oscillates, rotates, or varies periodically.
| View | What It Studies | Where It Appears |
|---|
| Triangle Trigonometry | Ratios of sides in right triangles; Law of Sines; Law of Cosines | Geometry, surveying, navigation, Chapter 7 |
| Function-Based Trigonometry | Sine, cosine, and related functions as functions of rotation; periodic behavior | Physics, engineering, calculus, Chapters 4–6, 13 |
The Central Transition
The shift from triangle trigonometry to function-based trigonometry is the main intellectual move of this chapter and the next several chapters. It requires extending the idea of angle from a static interior shape to a dynamic rotation, and introducing radian measure so that angles can be treated as real numbers.
MGU Library Connection
- For a review of right-triangle trigonometry from Geometry, see the Geometry chapter on Similar Triangles and Trigonometric Ratios.
- For connections to calculus applications, see Chapter 13 (Limits) and the Calculus I preview in Chapter 14.
Section 3.2
Angles as Rotations
In Geometry, an angle is formed by two rays that share an endpoint called the vertex. In trigonometry, we extend this to a dynamic definition: an angle is a rotation—the result of rotating one ray (the initial side) to a new position (the terminal side) around the vertex.
This definition allows angles to be larger than 360° (more than one full rotation), negative (clockwise rotation), or anywhere on the real number line.
Standard Position
An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. The terminal side rotates counterclockwise (positive) or clockwise (negative) from there.
Diagram Instruction
Draw a coordinate plane. Draw a ray from the origin along the positive x-axis and label it "initial side." Draw a second ray from the origin at a counterclockwise angle, label it "terminal side." Draw a curved arrow from the initial to terminal side and label the angle θ. Draw a second example showing a clockwise (negative) angle.
| Terminal Side Location | Quadrant or Axis |
|---|
| Between 0° and 90° | Quadrant I |
| Between 90° and 180° | Quadrant II |
| Between 180° and 270° | Quadrant III |
| Between 270° and 360° | Quadrant IV |
| Exactly 0°, 180°, 360° | On the x-axis (quadrantal) |
| Exactly 90°, 270° | On the y-axis (quadrantal) |
Section 3.3
Degree Measure Review
Degrees are the most familiar unit for measuring angles. One full rotation equals 360°. This convention traces to ancient Babylonian astronomy. When higher precision is needed, degrees are subdivided into minutes (') and seconds ('').
DMS Units
1° = 60′ 1′ = 60″ So 1° = 3600″
Example 3.3.1
Converting DMS to Decimal Degrees
- ProblemConvert 47° 24′ 36″ to decimal degrees.
- ThinkDivide minutes by 60 and seconds by 3600 to convert each to fractions of a degree, then add.
- Solve
47 + 24/60 + 36/3600 = 47 + 0.4 + 0.01 = 47.41°
- Check24 out of 60 minutes = 0.4°; 36 out of 3600 seconds = 0.01°. Reasonable. ✓
Answer: 47° 24′ 36″ = 47.41°
Example 3.3.2
Converting Decimal Degrees to DMS
- ProblemConvert 133.275° to DMS.
- ThinkWhole degrees: 133. Multiply the decimal part by 60 to get minutes. If there is a remaining decimal in minutes, multiply by 60 for seconds.
- Solve
0.275 × 60 = 16.5′
0.5′ × 60 = 30″
Answer: 133.275° = 133° 16′ 30″
Section 3.4
Radian Measure
Degrees work perfectly well for everyday angle measurement, but they are an arbitrary convention. There is no deep geometric reason why a full rotation equals 360°. In mathematics, especially in calculus and physics, it is far more convenient to measure angles using a unit that arises naturally from the geometry of circles. That unit is the radian.
Definition of a Radian
The radian measure of an angle is the ratio of the arc length s cut off by the angle to the radius r of the circle:
θ = s / r
When the arc length equals the radius — when s = r — the angle measures exactly 1 radian.
Diagram Instruction
Draw a circle of radius r centered at the origin. Draw two radii from the center: one along the positive x-axis (initial side) and one at a counterclockwise angle (terminal side). Label the arc between them s and the radius r. Show the special case where s = r and label that angle "1 radian."
Since the full circumference is 2πr, and each radian corresponds to an arc equal to r, you can wrap the radius exactly 2π times around the circle before returning to the start. Therefore one full rotation equals 2π radians.
The Key Insight
Radians measure angles by asking: "How many radii fit along the arc?" One radian means the arc is one radius long. This is a geometric, circle-based definition — and it makes radian measure essential once you study functions, calculus, and physics. In calculus, the derivatives of sin and cos only work out cleanly when the input is in radians.
| Degrees | Radians (Exact) | Description |
|---|
| 0° | 0 | No rotation |
| 30° | π/6 | One-twelfth of a full rotation |
| 45° | π/4 | One-eighth of a full rotation |
| 60° | π/3 | One-sixth of a full rotation |
| 90° | π/2 | Quarter turn |
| 120° | 2π/3 | One-third of a full rotation |
| 135° | 3π/4 | Three-eighths of a full rotation |
| 150° | 5π/6 | Five-twelfths |
| 180° | π | Half rotation |
| 270° | 3π/2 | Three-quarter rotation |
| 360° | 2π | Full rotation |
Section 3.5
Converting Between Degrees and Radians
Conversion Rules
Degrees → Radians: multiply by π/180
Radians → Degrees: multiply by 180/π
Memory device: to convert Degrees to radians, π goes over 180 (π on top, like π over a Desk).
Example 3.5.1
Converting Degrees to Radians
- ProblemConvert 240° to radians.
- Solve
240 × π/180 = 240π/180 = 4π/3
Answer: 240° = 4π/3 radians
Example 3.5.2
Converting a Negative Angle to Radians
- ProblemConvert −315° to radians.
- Solve
−315 × π/180 = −315π/180 = −7π/4
Answer: −315° = −7π/4 radians
Example 3.5.3
Converting Radians to Degrees
- ProblemConvert 5π/6 radians to degrees.
- Solve
(5π/6) × (180/π) = (5 × 180)/6 = 900/6 = 150°
Answer: 5π/6 radians = 150°
Example 3.5.4
Converting a Non-Standard Radian Value
- ProblemConvert 2.4 radians to degrees (round to the nearest tenth).
- ThinkWhen the radian measure is a decimal not involving π, the degree result will not be a "nice" number. Multiply by 180/π and use a decimal approximation.
- Solve
2.4 × (180/π) ≈ 2.4 × 57.296 ≈ 137.5°
Answer: 2.4 radians ≈ 137.5°
Two Common Conversion Mistakes
Mistake 1: Forgetting to simplify. 240π/180 must be reduced to 4π/3. Always simplify fractional radian expressions before using them.
Mistake 2: Confusing decimal radians with decimal degrees. The angle 2.4° is tiny (less than 3°). The angle 2.4 radians is about 137.5°. Always track the unit.
Convert to radians (give exact answers). For 9–10, convert to degrees.
- 30°
- 45°
- 120°
- 210°
- −150°
- 540°
- π/4
- 7π/6
- −5π/3
- 3.1 radians (approx.)
Section 3.6
Coterminal Angles
Two angles are coterminal if they have the same initial side and the same terminal side when placed in standard position — that is, they represent different amounts of rotation but end in the same position. Adding or subtracting any number of full rotations (360° or 2π) produces a coterminal angle.
Coterminal Angles
In degrees: θ + 360°k for any integer k
In radians: θ + 2πk for any integer k
Example 3.6.1
Coterminal Angles in Degrees
- ProblemFind a positive and a negative angle coterminal with 50°.
- Solve
50° + 360° = 410°
50° − 360° = −310°
Answer: 410° and −310° are both coterminal with 50°.
Example 3.6.2
Coterminal Angles in Radians
- ProblemFind a positive and a negative angle coterminal with 2π/3.
- Solve
2π/3 + 2π = 2π/3 + 6π/3 = 8π/3
2π/3 − 2π = 2π/3 − 6π/3 = −4π/3
Answer: 8π/3 and −4π/3 are coterminal with 2π/3.
Example 3.6.3
Finding the Coterminal Angle Between 0° and 360°
- ProblemFind the angle between 0° and 360° coterminal with −500°.
- ThinkAdd 360° repeatedly until the result falls in [0°, 360°).
- Solve
−500° + 360° = −140° (still negative)
−140° + 360° = 220° ✓
Answer: 220° is coterminal with −500°.
Section 3.7
Reference Angles
The reference angle for an angle θ in standard position is the positive acute angle between the terminal side of θ and the nearest part of the x-axis. Reference angles are always between 0° and 90° (or 0 and π/2 radians).
Diagram Instruction
Draw four coordinate planes, one for each quadrant. In each, draw an angle in standard position with its terminal side in the corresponding quadrant. Draw the reference angle θ′ as the acute angle between the terminal side and the nearest x-axis direction. Label both θ and θ′.
| Quadrant | Reference Angle (Degrees) | Reference Angle (Radians) |
|---|
| I | θ′ = θ | θ′ = θ |
| II | θ′ = 180° − θ | θ′ = π − θ |
| III | θ′ = θ − 180° | θ′ = θ − π |
| IV | θ′ = 360° − θ | θ′ = 2π − θ |
Example 3.7.1
Reference Angle in Degrees
- ProblemFind the reference angle for θ = 225°.
- Think225° is between 180° and 270°, so the terminal side is in Quadrant III.
- Solve
θ′ = 225° − 180° = 45°
Answer: The reference angle is 45°.
Example 3.7.2
Reference Angle in Radians
- ProblemFind the reference angle for θ = 5π/6.
- Think5π/6 is between π/2 and π, so the terminal side is in Quadrant II.
- Solve
θ′ = π − 5π/6 = 6π/6 − 5π/6 = π/6
Answer: The reference angle is π/6.
Example 3.7.3
Reference Angle for a Negative Angle
- ProblemFind the reference angle for θ = −130°.
- ThinkFirst find the coterminal angle between 0° and 360°; then find the reference angle for that angle.
- Solve
−130° + 360° = 230° (Quadrant III)
θ′ = 230° − 180° = 50°
Answer: The reference angle is 50°.
For 1–4, find a positive and negative coterminal angle. For 5–8, find the reference angle.
- 40°
- −80°
- 7π/4
- −2π/3
- 130°
- 205°
- 5π/7
- 11π/6
Sections 3.8 – 3.9
Arc Length and Sector Area
Arc Length Formula · Sector Area Formula · Applications and Problem Solving
Section 3.8
Arc Length
The definition of radian measure (θ = s/r) can be rearranged directly to give arc length. For a central angle θ (in radians) on a circle of radius r:
Diagram Instruction
Draw a circle of radius r. Draw a central angle θ at the center. Label the arc between the two radii as s. Show the equation s = rθ alongside the arc.
Example 3.8.1
Finding Arc Length (Angle in Radians)
- ProblemFind the arc length intercepted by a central angle of 3π/4 on a circle of radius 6 meters.
- ThinkThe angle is already in radians. Apply s = rθ directly.
- Solve
s = 6 · (3π/4) = 18π/4 = 9π/2 ≈ 14.14 m
Answer: s = 9π/2 ≈ 14.14 meters
Example 3.8.2
Finding Arc Length (Angle in Degrees)
- ProblemFind the arc length intercepted by a central angle of 120° on a circle of radius 9 cm.
- ThinkConvert 120° to radians first; then apply s = rθ.
- Solve
120° × π/180 = 2π/3 radians
s = 9 · (2π/3) = 18π/3 = 6π ≈ 18.85 cm
Answer: s = 6π ≈ 18.85 cm
Example 3.8.3
Finding the Radius from Arc Length
- ProblemAn arc of length 15 ft is intercepted by a central angle of 5/3 radians. Find the radius.
- Solve
s = rθ → 15 = r · (5/3) → r = 15 · (3/5) = 9 ft
Answer: r = 9 feet
Example 3.8.4
Finding the Central Angle from Arc Length
- ProblemA circle has radius 4 in. An arc of length π in is cut off. What is the central angle in radians? In degrees?
- Solve
θ = s/r = π/4 radians
(π/4) × (180/π) = 45°
Answer: θ = π/4 radians = 45°
Critical Reminder
The arc length formula s = rθ requires θ in radians. Using degrees without converting produces an answer off by a factor of π/180. This is one of the most common errors in this chapter.
Section 3.9
Sector Area
A sector of a circle is the region bounded by two radii and the arc between them — like a slice of pie. The sector takes up a fraction θ/(2π) of the full circle area πr², giving:
Example 3.9.1
Finding Sector Area
- ProblemFind the area of a sector with central angle π/3 and radius 6 cm.
- Solve
A = ½(6²)(π/3) = ½ · 36 · (π/3) = 36π/6 = 6π ≈ 18.85 cm²
Answer: A = 6π ≈ 18.85 cm²
Example 3.9.2
Sector Area from a Degree Angle
- ProblemFind the area of a sector with central angle 150° and radius 10 ft.
- ThinkConvert 150° to radians first.
- Solve
150° × π/180 = 5π/6 radians
A = ½(100)(5π/6) = 500π/12 = 125π/3 ≈ 130.9 ft²
Answer: A = 125π/3 ≈ 130.9 ft²
Example 3.9.3
Finding the Angle from Sector Area and Radius
- ProblemA sector has area 24π sq in and radius 12 in. Find the central angle in radians.
- Solve
24π = ½(144)θ = 72θ → θ = 24π/72 = π/3 radians
Answer: θ = π/3 radians
- s = ? when r = 8, θ = π/2
- s = ? when r = 12, θ = 3π/4
- s = ? when r = 10, θ = 120°
- r = ? when s = 10, θ = 2π/5
- θ = ? when s = 6π, r = 9
- A = ? when r = 4, θ = 3π/4
- A = ? when r = 9, θ = 2π/3
- A = ? when r = 6, θ = 150°
- r = ? when A = 50, θ = 5π/6
- Sector r = 12, θ = 7π/12: find both s and A.
Sections 3.10 – 3.12
Angular Speed, Linear Speed, and Applications
Angular Speed · Linear Speed · Periodic Motion · Navigation · Physics · Engineering
Section 3.10
Angular and Linear Speed
When a point moves along a circular path, there are two related but distinct rates. Angular speed (ω, omega) measures how fast the angle changes — radians per unit time. Linear speed (v) measures how fast the point travels along the arc — distance per unit time. The radius connects them.
Speed Formulas
ω = θ/t Angular speed (rad per unit time)
v = s/t Linear speed (distance per unit time)
v = rω Connecting them (since s = rθ, divide both sides by t)
A point farther from the center has a larger v even when ω is the same — this is why the edge of a disk moves faster than a point near the hub.
Example 3.10.1
Ferris Wheel — Angular and Linear Speed
- ProblemA Ferris wheel has radius 25 m and completes one revolution every 40 seconds. Find (a) angular speed and (b) the linear speed of a rider on the rim.
- ThinkOne full revolution = 2π radians.
- Solve
(a) ω = 2π/40 = π/20 rad/s ≈ 0.157 rad/s
(b) v = rω = 25 · (π/20) = 25π/20 = 5π/4 ≈ 3.93 m/s
Answer: ω = π/20 rad/s; v ≈ 3.93 m/s
Example 3.10.2
Converting RPM to Radians per Second
- ProblemAn electric motor spins at 1800 rpm. Find the angular speed in rad/s.
- ThinkEach revolution = 2π radians. Convert to rad/min, then divide by 60.
- Solve
1800 rev/min × 2π rad/rev = 3600π rad/min
ω = 3600π / 60 = 60π ≈ 188.5 rad/s
Answer: ω = 60π ≈ 188.5 rad/s
Example 3.10.3
Linear Speed of a Gear Rim
- ProblemA gear of radius 8 cm rotates at ω = 3π/2 rad/s. How fast is a point on the rim moving?
- Solve
v = rω = 8 · (3π/2) = 12π ≈ 37.7 cm/s
Answer: v = 12π ≈ 37.7 cm/s
Section 3.11
Circles and Periodic Motion
A motion is periodic if it repeats at regular intervals. Because angles repeat after every 2π radians, the position of any point on a rotating circle repeats after every full revolution. The time needed for one complete cycle is the period T. The number of cycles per unit time is the frequency f.
Period and Frequency
T = 2π / ω Period (time per revolution)
f = 1 / T = ω / (2π) Frequency (revolutions per unit time)
Diagram Instruction — Preview of Sine
Draw a unit circle. Mark a point P on the rim at angles 0, π/2, π, 3π/2, and 2π. Beside the circle, sketch a wavy graph showing the height (vertical position) of P as a function of angle — a sine curve. Label the height axis "y = sin θ." This connects circular motion to the trigonometric graphs developed in Chapter 5.
Example 3.11.1
Period and Frequency from Angular Speed
- ProblemA wheel rotates at ω = 4π rad/s. Find the period and frequency.
- Solve
T = 2π / 4π = 1/2 second per revolution
f = 1 / T = 2 rev/s (2 Hz)
Answer: T = ½ s; f = 2 Hz
Why This Matters for Later Chapters
The word "periodic" will appear throughout this course. Trigonometric graphs are periodic. Trigonometric equations have infinitely many solutions because they are periodic. Chapter 5 will show how amplitude, period, phase shift, and vertical shift can be read directly from a sinusoidal graph.
Section 3.12
Angles in Navigation, Physics, and Engineering
Navigation: Bearings
In navigation, directions are given using bearings: angles measured from north, rotating clockwise. A bearing of N 45° E means start from north and rotate 45° toward east. Bearings may also be given as three-digit numbers from 000° (north) to 359°, clockwise.
Diagram Instruction
Draw a compass rose with north (up), east (right), south (down), west (left). Draw a bearing of N 60° E as an arrow at 60° from north toward east. Draw S 30° W as a second arrow.
Example 3.12.1
Navigation with Bearings
- ProblemA ship travels N 60° E for 50 nautical miles. How far north and how far east has it traveled?
- RepresentThis is a right triangle. The hypotenuse is 50 nm; the angle from north is 60°. North component = 50 cos 60°; East component = 50 sin 60°.
- Solve
North = 50 cos 60° = 50 · ½ = 25 nm
East = 50 sin 60° = 50 · (√3/2) = 25√3 ≈ 43.3 nm
Answer: 25 nm north and ≈ 43.3 nm east.
Physics and Engineering
Radian measure appears in centripetal acceleration (a = v²/r = rω²), rotational kinetic energy (KE = ½Iω²), small-angle approximations in pendulum equations (sin θ ≈ θ in radians), and sinusoidal signal descriptions f(t) = A sin(ωt + φ). In every case, trying to use degrees instead of radians requires an additional conversion factor of π/180 that clutters every step.
Example 3.12.2
Angular Speed of a Clock's Minute Hand
- ProblemFind the angular speed of the minute hand of a clock in rad/s.
- ThinkThe minute hand completes 2π radians in 60 minutes = 3600 seconds.
- Solve
ω = 2π / 3600 = π/1800 ≈ 0.00175 rad/s
Answer: ω = π/1800 rad/s
Example 3.12.3
Angular Speed of a Bicycle Wheel
- ProblemA bicycle wheel has radius 0.35 m. The bicycle moves at 5 m/s. Find the angular speed of the wheel.
- ThinkThe linear speed of a rim point equals the bicycle's speed. Use v = rω and solve for ω.
- Solve
ω = v/r = 5 / 0.35 ≈ 14.3 rad/s
Answer: ω ≈ 14.3 rad/s
MGU Library Connection
- For a deeper look at sinusoidal signals and phase relationships, see the Engineering and Signal Processing notes in the MGU Library.
- For how the formula f(t) = A sin(ωt + φ) is developed, see Chapter 5 (Graphs of Trigonometric Functions).
- A wheel of radius 0.5 m rotates at ω = 8 rad/s. Find the linear speed of a rim point.
- An LP record spins at 33⅓ rpm. Find the angular speed in rad/s.
- A satellite orbits at r = 7000 km and completes one orbit in 100 minutes. Find the linear speed in km/min.
- A ceiling fan blade (length 0.7 m) completes 2 full revolutions per second. Find ω in rad/s and v (tip) in m/s.
- A gear with radius 3 cm drives a gear with radius 9 cm. The small gear rotates at 180° per second. Find the angular speed of the large gear in rad/s. (Hint: linear speeds at contact point are equal.)
- Find the period T and frequency f for a wheel rotating at ω = 6π rad/s.
- A ship travels on bearing S 45° W for 60 miles. How far south and how far west does it travel?
- Earth's radius is 6371 km. Two cities are separated by a central angle of π/6 radians along the same meridian. Find the distance between them along Earth's surface.
Sections 3.13 – 3.14
Common Mistakes and the Road to the Unit Circle
Eight Recurring Errors · Preparing for Chapter 4
Section 3.13
Common Angle Measure Mistakes
Mistake 1 — Using Arc Length Formula with Degrees
s = rθ with θ = 60°: s = 6 · 60 = 360
Correct: convert 60° = π/3 radians first. Then s = 6 · (π/3) = 2π. The formula s = rθ only works when θ is in radians.
Mistake 2 — Confusing sin⁻¹ with 1/sin
Treating sin⁻¹(x) as 1/sin(x)
Correct: sin⁻¹(x) is the inverse sine function, returning an angle whose sine is x. The reciprocal of sine is csc(x) = 1/sin(x). These are completely different.
Mistake 3 — Not Simplifying Radian Conversions
Leaving 240π/180 without simplifying
Correct: always reduce the fraction — 240π/180 = 4π/3. Unsimplified radian expressions are harder to compare and use in calculations.
Mistake 4 — Ignoring the Direction of Rotation
Treating −50° and 50° as the same angle
Correct: −50° rotates clockwise; 50° rotates counterclockwise. Their terminal sides are reflections of each other across the x-axis, not the same position.
Mistake 5 — Using Sector Area Formula with Degrees
A = ½r²θ with θ = 90°: A = ½(4²)(90) = 720
Correct: convert 90° = π/2 radians first. A = ½(16)(π/2) = 4π ≈ 12.57. Using degrees gives an answer off by a factor of π/180.
Mistake 6 — Inverting v = rω
Writing ω = r · v instead of ω = v/r
Correct: v = rω, so ω = v/r. Always write the formula first, then solve for the unknown. Check units before substituting.
Mistake 7 — Saying Coterminal Angles Are Equal
"410° and 50° are the same angle."
Correct: they are coterminal — they have the same terminal side — but 410° represents one full rotation plus 50°, while 50° is only 50°. They are different rotations that end in the same position.
Mistake 8 — Using 360° − θ for All Reference Angles
Always computing 360° − θ regardless of quadrant
Correct: use the quadrant-specific formula: QI: θ′ = θ · QII: θ′ = 180° − θ · QIII: θ′ = θ − 180° · QIV: θ′ = 360° − θ. The formula 360° − θ is only correct in QIV.
Section 3.14
Preparing for the Unit Circle
Chapter 4 builds the unit circle — a circle of radius 1 centered at the origin — as the foundation for all six trigonometric functions. Everything developed in this chapter provides the essential preparation for that work.
Standard position gives a consistent coordinate setting. Radian measure makes angles into real numbers so that sine and cosine can be defined as functions f: ℝ → ℝ. Coterminal angles explain periodicity. Reference angles reduce any evaluation to Quadrant I. And on the unit circle (r = 1), the arc length formula becomes s = 1 · θ = θ: the radian measure of an angle is numerically equal to the arc it intercepts. Angles and arc lengths on the unit circle are the same number.
Unit Circle Preview
On the unit circle (r = 1), the point P at angle θ has coordinates:
P = (cos θ, sin θ)
The x-coordinate defines cosine. The y-coordinate defines sine. This will be built carefully in Chapter 4.
| Angle θ | Point P on Unit Circle |
|---|
| θ = 0 | (1, 0) |
| θ = π/2 | (0, 1) |
| θ = π | (−1, 0) |
| θ = 3π/2 | (0, −1) |
| θ = 2π | back to (1, 0) |
Diagram Instruction — Unit Circle Preview
Draw a unit circle centered at the origin. Mark and label the four axis-intercept points: (1, 0) at θ = 0, (0, 1) at θ = π/2, (−1, 0) at θ = π, and (0, −1) at θ = 3π/2. Indicate that more special angle positions — at π/6, π/4, π/3, and their quadrant multiples — will be added in Chapter 4.
Bridge to Chapter 4
The angles you have practiced converting and classifying in this chapter — π/6, π/4, π/3, π/2, and their multiples — will become the key reference positions on the unit circle. Their exact coordinates will be derived from 30-60-90 and 45-45-90 triangle geometry and will serve as the core unit circle values used throughout the trigonometry sequence.
Chapter 3 — Back Matter
Chapter Summary
- Sec 3.1Trigonometry has two major forms: triangle trigonometry and function-based trigonometry rooted in circles and rotation. This book develops both, with the function-based view as the main path forward into calculus.
- Sec 3.2An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. Positive angles rotate counterclockwise; negative angles rotate clockwise. Angles can be greater than 360° or less than −360°. Angles on the axes are quadrantal.
- Sec 3.3One full rotation = 360°. Minutes and seconds subdivide degrees: 1° = 60′ and 1′ = 60″. Convert DMS to decimal degrees by dividing minutes by 60 and seconds by 3600 and adding to whole degrees.
- Sec 3.4One radian is the angle whose arc equals the radius of the circle (θ = s/r). One full rotation = 2π radians. The fundamental conversion is 180° = π radians. Radians are dimensionless and natural for all circle-based formulas.
- Sec 3.5To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π. Always simplify fractional radian answers. Decimal radian values are very different from the same decimal in degrees.
- Sec 3.6Coterminal angles have the same terminal side. Add or subtract multiples of 360° (or 2π) to find coterminals. To find the coterminal angle in [0°, 360°), add 360° until the result is non-negative and less than 360°.
- Sec 3.7The reference angle is the positive acute angle between the terminal side and the nearest x-axis. It is always in [0°, 90°]. Use quadrant-specific formulas: QI: θ′=θ, QII: θ′=180°−θ, QIII: θ′=θ−180°, QIV: θ′=360°−θ.
- Sec 3.8Arc length: s = rθ with θ in radians. This formula follows directly from the definition of a radian and requires no extra conversion factor. To use a degree angle, convert to radians first.
- Sec 3.9Sector area: A = ½r²θ with θ in radians. This is derived from the sector's fractional share of the full circle. Both arc length and sector area formulas require radian measure.
- Sec 3.10Angular speed ω = θ/t (rad per time). Linear speed v = s/t. They are connected by v = rω. A point farther from the center has greater linear speed for the same angular speed.
- Sec 3.11Circular motion is periodic because angles repeat every 2π radians. Period T = 2π/ω. Frequency f = 1/T. The height of a point on a rotating circle traces a sine curve — a preview of Chapter 5.
- Sec 3.12Bearings describe navigation directions as clockwise angles from north. Angular and linear speed appear throughout physics and engineering. Radian measure is essential in all physical formulas involving rotation — using degrees requires an extra π/180 factor everywhere.
- Sec 3.13Eight common mistakes: using arc/area formulas with degrees; confusing sin⁻¹ with 1/sin; unsimplified radian expressions; ignoring rotation direction; wrong reference angle formula for quadrant; inverting v = rω; calling coterminal angles equal; applying 360° − θ in all quadrants.
- Sec 3.14Chapter 4 defines sine and cosine as coordinates on the unit circle. On the unit circle (r = 1), arc length equals radian measure: s = θ. The angles mastered here — π/6, π/4, π/3, π/2, and their multiples — will become the special positions of the unit circle.
Key Terms Review
angle
A rotation from an initial side to a terminal side around a vertex. In standard position, the vertex is at the origin and the initial side is along the positive x-axis.
initial side
The fixed starting ray of an angle in standard position, placed along the positive x-axis.
terminal side
The ray that results after the rotation. Its position determines the quadrant of the angle.
standard position
An angle placement with vertex at the origin and initial side on the positive x-axis.
degree
A unit of angle measure in which one full rotation equals 360°. Subdivided into minutes (60′ = 1°) and seconds (60″ = 1′).
radian
A unit of angle measure defined by θ = s/r; the angle whose arc equals the radius. One full rotation equals 2π radians.
arc length
The length of an arc intercepted by a central angle: s = rθ when θ is in radians.
sector
The region of a circle bounded by two radii and the arc between them — like a pie slice.
sector area
The area of a sector: A = ½r²θ when θ is in radians.
coterminal angles
Angles in standard position with the same terminal side; they differ by multiples of 360° or 2π. Different rotations that end in the same position.
reference angle
The positive acute angle between the terminal side of an angle and the nearest portion of the x-axis. Always between 0° and 90°.
quadrantal angle
An angle whose terminal side lies exactly on one of the coordinate axes (0°, 90°, 180°, 270°, or their equivalents).
angular speed (ω)
The rate of change of angle with respect to time: ω = θ/t. Measured in radians per unit time.
linear speed (v)
The rate of travel along the arc: v = s/t. Connected to angular speed by v = rω.
period (T)
The time required for one complete cycle of a periodic motion: T = 2π/ω.
frequency (f)
The number of cycles per unit time: f = 1/T = ω/(2π). Measured in hertz (Hz) when time is in seconds.
periodic motion
Motion that repeats at regular intervals. Circular motion is periodic because angles repeat every 2π radians.
bearing
A navigation direction given as an angle measured clockwise from north (e.g., N 45° E or 045°).
unit circle
A circle of radius 1 centered at the origin. Every point P on the unit circle at angle θ has coordinates (cos θ, sin θ). Foundation for Chapter 4.
Concept Review Questions
- Explain the difference between a triangle angle and a rotation angle. Why does the rotation definition extend the meaning of angle usefully?
- Why is radian measure considered more natural than degree measure for advanced mathematics? Give at least two reasons based on formulas from this chapter.
- A full rotation is 2π radians. Explain why, using the definition of a radian.
- Explain in words what it means for two angles to be coterminal. Give an example in both degrees and radians.
- What is a reference angle? Why is the reference angle always between 0° and 90°?
- Why do the arc length and sector area formulas require radians rather than degrees?
- Explain the difference between angular speed and linear speed. Why do points at different radii on the same rotating disk have different linear speeds but the same angular speed?
- A clock's second, minute, and hour hands all rotate. Which has the greatest angular speed? How does this affect the linear speed of the tip of each hand?
- What does "periodic" mean? How does circular motion produce periodic behavior?
- Give an example from navigation, physics, or engineering where radian measure and angular speed appear naturally. Explain why degrees would be less convenient.
Degree and Radian Practice
Give exact answers in simplified form.
- 30°
- 45°
- 60°
- 90°
- 120°
- 150°
- 180°
- 210°
- 225°
- 240°
- 270°
- 300°
- 315°
- 330°
- 360°
- −45°
- −120°
- −270°
- 540°
- −315°
- π/6
- π/4
- π/3
- π/2
- 2π/3
- 3π/4
- 5π/6
- π
- 7π/6
- 5π/4
- 4π/3
- 3π/2
- 7π/4
- 2π
- −π/3
- −3π/2
- 3π
- −7π/4
- 2 radians (approx.)
- 5 radians (approx.)
For 1–7: find a positive and a negative coterminal angle.
- 40°
- 155°
- −80°
- −200°
- π/5
- 7π/4
- −2π/3
For 8–12: find the coterminal angle in [0°, 360°) or [0, 2π).
- −50°
- 730°
- −π/6
- 13π/4
- −11π/3
For 13–20: find the reference angle.
- 130°
- 205°
- 310°
- −70°
- 5π/7
- 7π/5
- 11π/6
- −2π/5
Use s = rθ with θ in radians. Convert degree angles first.
- r = 8 m, θ = π/2
- r = 12 cm, θ = 3π/4
- r = 10 in, θ = 120°
- r = 6 ft, θ = 45°
- s = π ft, θ = 2π/5 — find r
- s = 10 cm, r = 5 cm — find θ (rad and deg)
- s = 15π in, θ = 5π/6 — find r
- Circumference 24π, θ = 60° — find s
- r = 7 m, θ = 2 radians — find s
- r = 5, s = 20 — find θ in radians
- r = 4 m, θ = 3π/4
- r = 9 cm, θ = 2π/3
- r = 6 ft, θ = 150°
- r = 10 in, θ = 45°
- A = 18π, r = 6 — find θ
- A = 50, θ = 5π/6 — find r
- Pie slice: r = 5 in, θ = 40° — find area
- r = 12, θ = 7π/12 — find both s and A
- Two sectors, same circle: θ₁ = 60°, θ₂ = 120° — ratio of areas?
- Fan blade: length 0.6 m, sweep angle 3π/4 — find area swept
Applications
- Bicycle wheel: diameter 68 cm, speed 12 km/h. Find the angular speed in rad/s.
- Ferris wheel: radius 20 m, one revolution every 3 minutes. Find (a) angular speed in rad/min and (b) linear speed in m/min.
- Clock hands: (a) Find angular speed of the hour hand in rad/min. (b) If the minute hand is 10 cm long, find the linear speed of its tip in cm/min.
- Satellite: orbital radius 7000 km, one orbit in 100 minutes. Find linear speed in km/min.
- Irrigation arm: length 40 m, sweeps 5π/6 radians. Find arc length traced by the tip.
- Windmill blade: length 6 m, rotating at 20 rpm. Find linear speed of the tip in m/s.
- Vinyl record: 33⅓ rpm LP. Find angular speed in rad/s.
- Navigation: A ship travels N 30° W for 80 miles. How far north and how far west?
- Gear pair: small gear r = 3 cm, large gear r = 9 cm. Small gear rotates at 180°/s. Find the angular speed of the large gear in rad/s.
- Earth's surface: Earth radius ≈ 6371 km. A central angle of π/6 radians separates two cities along the same meridian. Find the distance between them.
Error Analysis
Each problem shows a student's work containing an error. Identify the error and give the correct solution.
Problem 1
Convert π/5 to degrees: (π/5) × (π/180) = π²/900 ≈ 0.011°
Identify and correct the error. (Hint: check which conversion factor was used.)
Problem 2
Arc length: r = 6 cm, θ = 30°. Student writes s = 6 × 30 = 180 cm.
Identify the error. What is the correct arc length?
Problem 3
Reference angle for θ = 310°: student computes 310° − 180° = 130°.
Which quadrant is 310° in? What is the correct reference angle formula and answer?
Problem 4
Sector area: r = 5 cm, θ = 90°. Student writes A = ½(25)(90) = 1125 cm².
Identify and correct the error.
Problem 5
Wheel r = 0.5 m, ω = 10 rad/s. Student writes: ω = r · v → 10 = 0.5v → v = 20 m/s.
Is the numerical answer correct? Is the algebraic method correct? Explain.
Problem 6
Coterminal angle in [0°, 360°) for −400°: student adds 360° once, getting −40°, and stops.
What went wrong? What is the correct coterminal angle?
Chapter 3 Checkpoint
Chapter 3 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 3
Part A — Radian Measure and Conversions
- Define radian measure in your own words. Why is the radian a natural unit for measuring angles?
- Convert 225° to radians (exact).
- Convert −11π/6 to degrees.
- Convert 3 radians to degrees (round to the nearest tenth).
- Convert 47° 30′ to decimal degrees.
Part B — Coterminal and Reference Angles
- Find a positive and a negative coterminal angle for 7π/5.
- Find the coterminal angle in [0, 2π) for θ = −7π/3.
- Find the reference angle for θ = 7π/6.
- Find the reference angle for θ = −240°.
Part C — Arc Length and Sector Area
- Find the arc length intercepted by a central angle of 2π/3 on a circle with radius 9 cm.
- Find the arc length intercepted by a central angle of 135° on a circle with radius 8 ft.
- Find the area of a sector with radius 5 m and central angle 3π/4.
- A sector has area 27π in² and central angle 3π/4 radians. Find the radius.
Part D — Angular and Linear Speed
- A wheel rotates at π/3 radians per second. Find the period T (time for one full revolution).
- A disk with radius 0.4 m rotates at 5 rad/s. Find the linear speed of a point on the rim.
- A ceiling fan with blades of length 0.7 m completes 2 full revolutions per second. Find ω in rad/s and the linear speed of a blade tip in m/s.
Part E — Applications and Reasoning
- A radar antenna sweeps 5π/4 radians in 2 seconds. Find the angular speed.
- Two cities are on the same meridian, separated by a central angle of π/9 radians from Earth's center. Earth's radius ≈ 6371 km. Find the distance along Earth's surface between the cities.
- A circular sector has central angle 72° and arc length 8π inches. Find the radius and the sector area.
- Explain in writing how radian measure and arc length are related. Why does s = rθ require no extra constants when θ is in radians, while it would need a conversion factor if θ were in degrees?
Bridge Note to Chapter 4
Chapter 4: The Unit Circle and Trigonometric Functions
Chapter 3 built the foundation for everything that follows in the trigonometry sequence: standard position, radian measure, coterminal and reference angles, arc length, and periodic motion. Chapter 4 will use all of this to build the unit circle — a circle of radius 1 centered at the origin — as the foundation for trigonometric functions.
On the unit circle, every angle θ in standard position corresponds to a unique point P = (cos θ, sin θ). Because r = 1, the arc length formula becomes s = θ: the radian measure of an angle is numerically equal to the arc length on the unit circle. This is why radian measure is so efficient.
The angles practiced in this chapter — π/6, π/4, π/3, π/2, and their multiples through all four quadrants — will become the key reference positions of the unit circle. Their exact coordinates will be derived from 30-60-90 and 45-45-90 triangle geometry, and those exact values will be used throughout the trigonometry sequence, into identities, equations, graphs, and Calculus I.
MGU Library Connection
- Appendix F: Unit Circle Reference — complete labeled diagram with all special angles in degrees and radians
- Appendix G: Trigonometric Identities Reference — identities that build on the unit circle values developed in Chapter 4
- Appendix I: Law of Sines and Law of Cosines Reference — triangle applications returned to in Chapter 7
- Geometry chapter on Similar Triangles and Trigonometric Ratios — right-triangle review for the transition into function-based trigonometry
- MGU Physics I notes on Circular Motion and Simple Harmonic Motion — applications of angular speed, linear speed, and periodic motion