Trigonometry Diagrams

Reciprocal pair, frozen

Trigonometric Identities › Reciprocal Identities

A complementary pair, frozen at 35° and 55°

Trigonometric Identities › Co-Function Identities

Two periods, side by side

Trigonometric Identities › Periodicity Identities

Half-angle derivation, final step

Trigonometric Identities › Half Angle Identities

Geometric proof, final step

Trigonometric Identities › Double Angle Identities

The arc and the radius, same length

Degrees and Radians › Radian Measurement

Radius, angle, arc

Degrees and Radians › Arc Length

The slice against the whole

Degrees and Radians › Sector Area

Standard position

Degrees and Radians › Standard Position of an Angle

Coterminal pair, frozen

Degrees and Radians › Coterminal Angles

Complementary 35° + 55°; supplementary 110° + 70°

Degrees and Radians › Complementary and Supplementary Angles

y = sin x repeats every 2π

Trigonometric Equations › General Solutions vs Restricted Solutions

sin 30° = sin 150° = 1/2

Trigonometric Equations › Solving Basic Sine Equations

y = tan x: one crossing per branch

Trigonometric Equations › Solving Basic Tangent Equations

Same line, same interval, twice the crossings

Trigonometric Equations › Equations with Multiple Angles

sin 50° ≈ 0.766, the vertical leg

Trigonometric Functions › The Sine Function

cos 50° ≈ 0.643, the horizontal leg

Trigonometric Functions › The Cosine Function

tan 50° ≈ 1.192, the slope of the radius

Trigonometric Functions › The Tangent Function

csc 30° = 2, hypotenuse over opposite

Trigonometric Functions › The Cosecant Function

sec 60° = 2, hypotenuse over adjacent

Trigonometric Functions › The Secant Function

cot 45° = 1, adjacent over opposite

Trigonometric Functions › The Cotangent Function

Four functions, two roots

Trigonometric Functions › Reciprocal and Quotient Relationships

Six functions, two patterns

Trigonometric Functions › Domain and Range Summary

From one value to all six

Trigonometric Functions › Finding All Function Values from One Known Value

y = sin x, marker at 60°

Trigonometric Graphs › Graph of the Sine Function

y = cos x, marker at 60°

Trigonometric Graphs › Graph of the Cosine Function

y = tan x, asymptotes π apart

Trigonometric Graphs › Graph of the Tangent Function

y = csc x, y = sec x, y = cot x at 60°

Trigonometric Graphs › Graphs of Cosecant, Secant, and Cotangent

One wave, four parameters

Trigonometric Graphs › The General Sinusoidal Form

A = 1 against A = 3

Trigonometric Graphs › Amplitude

B = 1 against B = 2

Trigonometric Graphs › Period

Where the cycle starts

Trigonometric Graphs › Phase Shift

D = 0 against D = 2

Trigonometric Graphs › Vertical Shift and Midline

Five readings, in order

Trigonometric Graphs › Determining the Equation from a Graph

One period, four equal parts

Trigonometric Graphs › Graphing by Hand: The Key-Point Method

Two points, or everything between them

Trigonometric Inequalities › Solutions as Intervals

y = sin x: where the curve sits above a level

Trigonometric Inequalities › The Graphical Method

Arc on the left, band on the right

Trigonometric Inequalities › The Unit Circle Method

Two bounds, one answer

Trigonometric Inequalities › Compound Inequalities

y = sec x: asymptotes at every zero of cosine

Trigonometric Inequalities › Domain Considerations for Reciprocal and Quotient Functions

One line, four crossings

Inverse Trigonometric Functions › Why Restriction Is Necessary

Sine and arcsine across y = x

Inverse Trigonometric Functions › The Arcsine Function

Cosine and arccosine across y = x

Inverse Trigonometric Functions › The Arccosine Function

Tangent and arctangent across y = x

Inverse Trigonometric Functions › The Arctangent Function

5π/6 in, π/6 out

Inverse Trigonometric Functions › Compositions of Trigonometric and Inverse Trigonometric Functions

What each inverse takes and gives

Inverse Trigonometric Functions › Graphs of Inverse Trigonometric Functions

One period of y = sin x is 2π

Trigonometric Properties › Periodicity

sin(−θ) = −sin θ: P reflected to P′

Trigonometric Properties › Even and Odd Symmetry

Inside the strip, and never inside it

Trigonometric Properties › Boundedness

θ = 180°: the point is (−1, 0)

Trigonometric Properties › Zeros

y = tan x: a break at every odd multiple of π/2

Trigonometric Properties › Continuity and Discontinuities

One interval each, chosen for a reason

Trigonometric Properties › Monotonicity on Principal Intervals

The same triangle, the other angle chosen

Right Triangle Trigonometry › Naming the Sides: Opposite, Adjacent, Hypotenuse

The 3-4-5 triangle: acute angles 36.9° and 53.1°

Right Triangle Trigonometry › Defining Sine, Cosine, and Tangent

One ratio per pair of sides

Right Triangle Trigonometry › Finding Missing Sides

Opposite 5, hypotenuse 13: θ = 22.6°

Right Triangle Trigonometry › Finding Missing Angles

45-45-90: two equal legs, hypotenuse √2 times a leg

Right Triangle Trigonometry › The 45-45-90 Triangle

30-60-90: sides in the ratio 1 : √3 : 2

Right Triangle Trigonometry › The 30-60-90 Triangle

Same leg, two names, two functions

Right Triangle Trigonometry › Cofunction Relationships

One line of sight, two observers

Right Triangle Trigonometry › Angles of Elevation and Depression

What happens at the two ends

Right Triangle Trigonometry › Limitations and Extensions

Law of Sines: a/sin A = b/sin B = c/sin C

Law of Sines and Cosines › The Law of Sines

The same b and A, three lengths of a

Law of Sines and Cosines › The Ambiguous Case (SSA)

Law of Cosines: c² = a² + b² − 2ab cos C

Law of Sines and Cosines › The Law of Cosines

The height, written with a sine

Law of Sines and Cosines › Area of a Triangle Using Trigonometry

The same angle on two circles

Unit Circle: Coordinates and Exact Values › Definition and Equation

θ = 50°: the point is (cos θ, sin θ)

Unit Circle: Coordinates and Exact Values › Coordinates as Trigonometric Values

Two functions break at each axis

Unit Circle: Coordinates and Exact Values › Standard Position and the Terminal Side

Quadrant II: x < 0, y > 0

Unit Circle: Coordinates and Exact Values › The Four Quadrants and Sign Patterns

140° with its reference angle of 40°

Unit Circle: Coordinates and Exact Values › Reference Angles

45° and −315°: one terminal side

Unit Circle: Coordinates and Exact Values › Angles Beyond One Rotation and Negative Angles

y = sin θ, marker at θ = 60°

Unit Circle: Coordinates and Exact Values › From Circular Motion to Waves

Acute angle, frozen at 50°

Angle Explorer › Acute Angles

Right angle, frozen at 90°

Angle Explorer › Right Angles

Obtuse angle, frozen

Angle Explorer › Obtuse Angles

Straight angle, frozen at 180°

Angle Explorer › Straight Angles

Reflex angle, frozen

Angle Explorer › Reflex Angles

Quadrant I, frozen

Angle Explorer › Quadrant I

Quadrant II, frozen

Angle Explorer › Quadrant II

Quadrant III, frozen

Angle Explorer › Quadrant III

Quadrant IV, frozen

Angle Explorer › Quadrant IV

0°, frozen

Angle Explorer › Special Angle: 0°

30°, frozen

Angle Explorer › Special Angle: 30°

45°, frozen

Angle Explorer › Special Angle: 45°

60°, frozen

Angle Explorer › Special Angle: 60°

90°, frozen

Angle Explorer › Special Angle: 90°

120°, frozen

Angle Explorer › Special Angle: 120°

135°, frozen

Angle Explorer › Special Angle: 135°

150°, frozen

Angle Explorer › Special Angle: 150°

180°, frozen

Angle Explorer › Special Angle: 180°

210°, frozen

Angle Explorer › Special Angle: 210°

225°, frozen

Angle Explorer › Special Angle: 225°

240°, frozen

Angle Explorer › Special Angle: 240°

270°, frozen

Angle Explorer › Special Angle: 270°

300°, frozen

Angle Explorer › Special Angle: 300°

315°, frozen

Angle Explorer › Special Angle: 315°

330°, frozen

Angle Explorer › Special Angle: 330°

360°, frozen

Angle Explorer › Special Angle: 360°

Complementary pair, frozen

Angle Explorer › Complementary Angles

Supplementary pair, frozen

Angle Explorer › Supplementary Angles

Reference angle, frozen at 140°

Angle Explorer › Reference Angles

Coterminal pair, frozen

Angle Explorer › Coterminal Angles

The six functions, frozen

Angle Explorer › The Six Trigonometric Functions

Complement and supplement, frozen

Interactive Angle Types Explorer › Complementary and Supplementary Angles

Vertical angles, frozen

Interactive Angle Types Explorer › Vertical Angles

Adjacent angles, frozen

Interactive Angle Types Explorer › Adjacent Angles

Standard position, frozen

Interactive Angle Types Explorer › Standard Position

Reference angle, frozen

Interactive Angle Types Explorer › Reference Angles

Coterminal pair, frozen

Interactive Angle Types Explorer › Coterminal Angles

The special angles, frozen

Interactive Angle Types Explorer › Special Angles on the Unit Circle

Directed angles, frozen

Interactive Angle Types Explorer › Directed Angles

Zero angle, frozen

Interactive Angle Types Explorer › Zero Angles

Acute angle, frozen

Interactive Angle Types Explorer › Acute Angles

Right angle, frozen at 90°

Interactive Angle Types Explorer › Right Angles

Obtuse angle, frozen

Interactive Angle Types Explorer › Obtuse Angles

Straight angle, frozen at 180°

Interactive Angle Types Explorer › Straight Angles

Reflex angle, frozen

Interactive Angle Types Explorer › Reflex Angles

Full angle, frozen at 360°

Interactive Angle Types Explorer › Full Angles

Complementary pair, frozen

Interactive Angle Types Explorer › Complementary Angles

Supplementary pair, frozen

Interactive Angle Types Explorer › Supplementary Angles

Standard position, Quadrant I

Interactive Angle Types Explorer › Quadrant I: Function Signs

Standard position, Quadrant II

Interactive Angle Types Explorer › Quadrant II: Function Signs

Standard position, Quadrant III

Interactive Angle Types Explorer › Quadrant III: Function Signs

Standard position, Quadrant IV

Interactive Angle Types Explorer › Quadrant IV: Function Signs

Reference angle, Quadrant I

Interactive Angle Types Explorer › Reference Angle in Quadrant I

Reference angle, Quadrant II

Interactive Angle Types Explorer › Reference Angle in Quadrant II

Reference angle, Quadrant III

Interactive Angle Types Explorer › Reference Angle in Quadrant III

Reference angle, Quadrant IV

Interactive Angle Types Explorer › Reference Angle in Quadrant IV

0°, frozen

Interactive Angle Types Explorer › Special Angle: 0°

30°, frozen

Interactive Angle Types Explorer › Special Angle: 30°

45°, frozen

Interactive Angle Types Explorer › Special Angle: 45°

60°, frozen

Interactive Angle Types Explorer › Special Angle: 60°

90°, frozen

Interactive Angle Types Explorer › Special Angle: 90°

120°, frozen

Interactive Angle Types Explorer › Special Angle: 120°

135°, frozen

Interactive Angle Types Explorer › Special Angle: 135°

150°, frozen

Interactive Angle Types Explorer › Special Angle: 150°

180°, frozen

Interactive Angle Types Explorer › Special Angle: 180°

210°, frozen

Interactive Angle Types Explorer › Special Angle: 210°

225°, frozen

Interactive Angle Types Explorer › Special Angle: 225°

240°, frozen

Interactive Angle Types Explorer › Special Angle: 240°

270°, frozen

Interactive Angle Types Explorer › Special Angle: 270°

300°, frozen

Interactive Angle Types Explorer › Special Angle: 300°

315°, frozen

Interactive Angle Types Explorer › Special Angle: 315°

330°, frozen

Interactive Angle Types Explorer › Special Angle: 330°

Positive angle, frozen

Interactive Angle Types Explorer › Positive Angles

Negative angle, frozen

Interactive Angle Types Explorer › Negative Angles

θ = 1 rad on a circle of radius 2

Arc and Sector Explorer › One Radian

θ = 3π/5, r = 2

Arc and Sector Explorer › Arc Length

θ = 2π/5, r = 2, sector shown as a fraction

Arc and Sector Explorer › Sector Area

θ = 2π/9 on the unit circle

Arc and Sector Explorer › Radius 1

Sine paired with its curve, frozen at 50°

Basic Trigonometric Identities Explorer › Sine on the Unit Circle

Cosine paired with its curve, frozen at 50°

Basic Trigonometric Identities Explorer › Cosine on the Unit Circle

Tangent paired with its curve, frozen at 50°

Basic Trigonometric Identities Explorer › Tangent on the Unit Circle

Cosecant paired with its curve, frozen at 50°

Basic Trigonometric Identities Explorer › Cosecant on the Unit Circle

Secant paired with its curve, frozen at 50°

Basic Trigonometric Identities Explorer › Secant on the Unit Circle

Cotangent paired with its curve, frozen at 50°

Basic Trigonometric Identities Explorer › Cotangent on the Unit Circle

Sine — Step 1: the angle placed

Basic Trigonometric Identities Explorer › Sine, Step 1: Place the Angle

Sine — Step 2: the leg highlighted

Basic Trigonometric Identities Explorer › Sine, Step 2: Drop the Vertical Leg

Sine — Step 3: pinned to the graph

Basic Trigonometric Identities Explorer › Sine, Step 3: Trace on the Graph

Sine — Step 4: reference angle at 140°

Basic Trigonometric Identities Explorer › Sine, Step 4: Reference Angle

Sine — Step 5: one turn later

Basic Trigonometric Identities Explorer › Sine, Step 5: Periodicity

Cosine — Step 1: the angle placed

Basic Trigonometric Identities Explorer › Cosine, Step 1: Place the Angle

Cosine — Step 2: the leg highlighted

Basic Trigonometric Identities Explorer › Cosine, Step 2: Project onto the X-Axis

Cosine — Step 3: pinned to the graph

Basic Trigonometric Identities Explorer › Cosine, Step 3: Trace on the Graph

Cosine — Step 4: reference angle at 140°

Basic Trigonometric Identities Explorer › Cosine, Step 4: Reference Angle

Cosine — Step 5: one turn later

Basic Trigonometric Identities Explorer › Cosine, Step 5: Periodicity

Tangent — Step 1: the angle placed

Basic Trigonometric Identities Explorer › Tangent, Step 1: Place the Angle

Tangent — Step 2: the leg highlighted

Basic Trigonometric Identities Explorer › Tangent, Step 2: Read Both Legs

Tangent — Step 3: pinned to the graph

Basic Trigonometric Identities Explorer › Tangent, Step 3: Form the Ratio

Tangent — Step 4: reference angle at 140°

Basic Trigonometric Identities Explorer › Tangent, Step 4: Sign by Quadrant

Tangent — Step 5: one turn later

Basic Trigonometric Identities Explorer › Tangent, Step 5: Periodicity

Cosecant — Step 1: the angle placed

Basic Trigonometric Identities Explorer › Cosecant, Step 1: Place the Angle

Cosecant — Step 2: the leg highlighted

Basic Trigonometric Identities Explorer › Cosecant, Step 2: Identify the Vertical Leg

Cosecant — Step 3: pinned to the graph

Basic Trigonometric Identities Explorer › Cosecant, Step 3: Take the Reciprocal

Cosecant — Step 4: reference angle at 140°

Basic Trigonometric Identities Explorer › Cosecant, Step 4: Range and Sign

Cosecant — Step 5: one turn later

Basic Trigonometric Identities Explorer › Cosecant, Step 5: Periodicity

Secant — Step 1: the angle placed

Basic Trigonometric Identities Explorer › Secant, Step 1: Place the Angle

Secant — Step 2: the leg highlighted

Basic Trigonometric Identities Explorer › Secant, Step 2: Identify the Horizontal Leg

Secant — Step 3: pinned to the graph

Basic Trigonometric Identities Explorer › Secant, Step 3: Take the Reciprocal

Secant — Step 4: reference angle at 140°

Basic Trigonometric Identities Explorer › Secant, Step 4: Range and Sign

Secant — Step 5: one turn later

Basic Trigonometric Identities Explorer › Secant, Step 5: Periodicity

Cotangent — Step 1: the angle placed

Basic Trigonometric Identities Explorer › Cotangent, Step 1: Place the Angle

Cotangent — Step 2: the leg highlighted

Basic Trigonometric Identities Explorer › Cotangent, Step 2: Read Both Legs

Cotangent — Step 3: pinned to the graph

Basic Trigonometric Identities Explorer › Cotangent, Step 3: Form the Ratio

Cotangent — Step 4: reference angle at 140°

Basic Trigonometric Identities Explorer › Cotangent, Step 4: Sign by Quadrant

Cotangent — Step 5: one turn later

Basic Trigonometric Identities Explorer › Cotangent, Step 5: Periodicity

The complete sine proof, frozen

Double Angle Identities Explorer › The Sine Double-Angle Identity

The complete cosine proof, frozen

Double Angle Identities Explorer › The Cosine Double-Angle Identity

tan(2θ), derived

Double Angle Identities Explorer › The Tangent Double-Angle Identity

csc(2θ), derived

Double Angle Identities Explorer › The Cosecant Double-Angle Identity

sec(2θ), derived

Double Angle Identities Explorer › The Secant Double-Angle Identity

cot(2θ), derived

Double Angle Identities Explorer › The Cotangent Double-Angle Identity

Step 1: the isosceles setup

Double Angle Identities Explorer › Sine Proof, Step 1: Setup

Step 2: area, first way

Double Angle Identities Explorer › Sine Proof, Step 2: Area, First Way

Step 3: the bisector

Double Angle Identities Explorer › Sine Proof, Step 3: Bisect

Step 4: legs as ratios

Double Angle Identities Explorer › Sine Proof, Step 4: Read Off the Legs

Step 5: area, second way

Double Angle Identities Explorer › Sine Proof, Step 5: Area, Second Way

Step 6: equate

Double Angle Identities Explorer › Sine Proof, Step 6: Equate

Step 1: same setup, new target

Double Angle Identities Explorer › Cosine Proof, Step 1: Setup

Step 2: law of cosines

Double Angle Identities Explorer › Cosine Proof, Step 2: Law of Cosines on Triangle OAB

Step 3: the bisector

Double Angle Identities Explorer › Cosine Proof, Step 3: Bisect

Step 4: the half-chord

Double Angle Identities Explorer › Cosine Proof, Step 4: Read Off the Half-Chord

Step 5: square it

Double Angle Identities Explorer › Cosine Proof, Step 5: Square the Chord

Step 6: equate

Double Angle Identities Explorer › Cosine Proof, Step 6: Equate

The complete sine derivation, frozen

Half Angle Identities Explorer › The Sine Half-Angle Identity

The complete cosine derivation, frozen

Half Angle Identities Explorer › The Cosine Half-Angle Identity

tan(α/2), derived

Half Angle Identities Explorer › The Tangent Half-Angle Identity

csc(α/2), derived

Half Angle Identities Explorer › The Cosecant Half-Angle Identity

sec(α/2), derived

Half Angle Identities Explorer › The Secant Half-Angle Identity

cot(α/2), derived

Half Angle Identities Explorer › The Cotangent Half-Angle Identity

Step 1: the setup

Half Angle Identities Explorer › Sine Half-Angle, Step 1: Setup

Step 2: law of cosines

Half Angle Identities Explorer › Sine Half-Angle, Step 2: Law of Cosines on Triangle OAB

Step 3: the bisector

Half Angle Identities Explorer › Sine Half-Angle, Step 3: Bisect

Step 4: the half-chord

Half Angle Identities Explorer › Sine Half-Angle, Step 4: Read Off the Half-Chord

Step 5: square the chord

Half Angle Identities Explorer › Sine Half-Angle, Step 5: Square the Chord

Step 6: equate and solve

Half Angle Identities Explorer › Sine Half-Angle, Step 6: Equate and Solve

Step 1: the setup

Half Angle Identities Explorer › Cosine Half-Angle, Step 1: Setup

Step 2: the bisector

Half Angle Identities Explorer › Cosine Half-Angle, Step 2: Bisect

Step 3: the legs

Half Angle Identities Explorer › Cosine Half-Angle, Step 3: Identify the Legs

Step 4: Pythagoras

Half Angle Identities Explorer › Cosine Half-Angle, Step 4: Apply Pythagoras

Step 5: substitute

Half Angle Identities Explorer › Cosine Half-Angle, Step 5: Substitute the Sin Half-Angle

Step 6: take the root

Half Angle Identities Explorer › Cosine Half-Angle, Step 6: Take the Root

The complete sine proof, frozen

Pythagorean Identities Explorer › The Sine Pythagorean Identity

The complete cosine proof, frozen

Pythagorean Identities Explorer › The Cosine Pythagorean Identity

tan θ, derived

Pythagorean Identities Explorer › The Tangent Pythagorean Identity

csc θ, derived

Pythagorean Identities Explorer › The Cosecant Pythagorean Identity

sec θ, derived

Pythagorean Identities Explorer › The Secant Pythagorean Identity

cot θ, derived

Pythagorean Identities Explorer › The Cotangent Pythagorean Identity

Step 1: the unit setup

Pythagorean Identities Explorer › Sine Proof, Step 1: Setup

Step 2: bisect

Pythagorean Identities Explorer › Sine Proof, Step 2: Bisect

Step 3: the legs named

Pythagorean Identities Explorer › Sine Proof, Step 3: Identify the Legs

Step 4: Pythagoras

Pythagorean Identities Explorer › Sine Proof, Step 4: Pythagoras

Step 5: solve for sin²θ

Pythagorean Identities Explorer › Sine Proof, Step 5: Solve for sin²θ

Step 6: the positive root

Pythagorean Identities Explorer › Sine Proof, Step 6: Take the Positive Root

Step 1: same setup, other leg

Pythagorean Identities Explorer › Cosine Proof, Step 1: Setup

Step 2: bisect

Pythagorean Identities Explorer › Cosine Proof, Step 2: Bisect