Chapter 10 — Graphs of Sine and Cosine

10.1 Learning Objectives

By the end of this chapter, you should be able to:

• graph sine and cosine functions
• understand periodic behavior
• identify amplitude
• identify period
• identify midline
• recognize wave patterns
• connect circular motion to graph behavior
• understand oscillation mathematically
• apply trig graphs to real-world systems
• prepare for advanced wave analysis and calculus

10.2 Big Picture — Trigonometry Becomes Wave Mathematics

Earlier chapters introduced:

• triangles
• trig ratios
• radians
• the unit circle

Now trigonometry transforms into:

• wave mathematics

Sine and cosine are not merely:

• triangle functions

They are:

• oscillating wave functions

These wave patterns appear throughout reality:

• sound
• light
• electricity
• tides
• heartbeats
• radio signals
• vibrations
• AI signal systems

Trig graphs become one of the great mathematical models of:

• periodic behavior

10.3 What Is a Periodic Function?

A periodic function:

• repeats in cycles

Examples:

• ocean tides
• seasons
• clock motion
• rotating wheels
• sound waves
• electrical current

Trig functions naturally model:

• repeating systems

10.4 The Sine Function

Basic sine function:

y = sin(x)

This graph oscillates smoothly between:

-1 and 1

The sine graph repeats forever.

10.5 The Cosine Function

Basic cosine function:

y = cos(x)

Cosine also oscillates between:

-1 and 1

Sine and cosine have similar wave shapes but begin at different positions.

10.6 Connection to the Unit Circle

The sine graph tracks:

y-coordinate

on the unit circle.

The cosine graph tracks:

x-coordinate

As the angle rotates:

coordinates oscillate

This creates wave motion naturally.

10.7 Important Sine Values

Angle sin(x)

0 0

π/2 1

π 0

3π/2 -1

2π 0

These points define the sine wave.

10.8 Important Cosine Values

Angle cos(x)

0 1

π/2 0

π -1

3π/2 0

2π 1

These points define the cosine wave.

10.9 Amplitude

Amplitude measures:

wave height

For:

y = sin(x)

amplitude equals:

1

General form:

y = A sin(x)

Amplitude equals:

|A|

10.10 Period

Period measures:

cycle length

For sine and cosine:

period = 2π

After:

2π

the graph repeats.

10.11 Midline

The midline represents:

wave center

For basic sine and cosine:

y = 0

Waves oscillate above and below the midline.

10.12 Frequency

Frequency measures:

how rapidly cycles repeat

Higher frequency:

more oscillations

Examples:

musical pitch

radio signals

sound waves

Trig graphs naturally model frequency systems.

10.13 Vertical Stretching

Example:

y = 3sin(x)

Amplitude becomes:

3

The wave becomes:

taller

10.14 Horizontal Stretching

Example:

y = sin(x/2)

Period becomes:

4π

The wave becomes:

wider

10.15 Phase Shift

Example:

y = sin(x - π/2)

The graph shifts horizontally.

Phase shifts model:

delayed systems

signal timing

wave synchronization

10.16 Real-World Waves

Trig graphs model:

sound

light

vibration

alternating current

radio transmission

seismic activity

Wave mathematics became foundational in modern science.

10.17 Sound Waves

Musical sound behaves approximately like:

sine waves

Higher frequency:

higher pitch

Larger amplitude:

louder sound

Trig graphs naturally describe acoustic systems.

10.18 Electricity and Alternating Current

Alternating current behaves like:

oscillating sine waves

Electrical engineering depends heavily on:

trig graphs

Power systems rely on periodic oscillation.

10.19 Trig Graphs and Physics

Physics constantly studies:

oscillation

harmonic motion

resonance

vibration

Trig graphs model these systems beautifully.

10.20 Trig Graphs and Computing

Computing applications include:

audio processing

graphics

AI signal analysis

image compression

communications systems

Wave mathematics powers much modern technology.

10.21 Visualization Matters

Trig graphs should be:

visualized dynamically

Students should:

sketch waves

mark peaks

mark troughs

identify repeating structure

Visualization builds intuition for periodic systems.

10.22 Common Beginner Difficulties

Students often struggle with:

radians on graphs

amplitude

period

phase shifts

visualizing oscillation

These struggles are normal.

Wave intuition develops through:

graph sketching

repetition

visualization

pattern recognition

10.23 Mental Model

Sine and cosine graphs model:

repeating oscillatory behavior

They connect:

circles

waves

rotation

vibration

periodic systems

Trig graphs become the mathematics of repeating motion.

10.24 Warm-Up Problems

Problems

Define periodic function.

Define amplitude.

Define period.

Define midline.

What is amplitude of:

y = sin(x)

What is period of:

y = sin(x)

What is amplitude of:

y = 4cos(x)

What is midline of:

y = sin(x)

Explain why sine graphs repeat.

Explain why cosine graphs repeat.

Explain why waves connect naturally to trig.

Explain why periodic systems matter.

10.25 Guided Problems

Problems

Find amplitude:

y = 5sin(x)

Find amplitude:

y = -3cos(x)

Find period:

y = sin(x)

Find period:

y = cos(x)

Evaluate:

sin(0)

Evaluate:

sin(π/2)

Evaluate:

cos(0)

Evaluate:

cos(π)

Explain why sound behaves like waves.

Explain why electricity uses oscillation.

Describe a real-world periodic system.

Explain why trig graphs became important in science.

10.26 Challenge Problems

Sketch key points for:

y = 2sin(x)

Sketch key points for:

y = 3cos(x)

Determine amplitude and period:

y = 7sin(x)

Determine amplitude and period:

y = -2cos(x)

Explain why circular motion creates wave motion.

Explain why periodic systems appear throughout nature.

Describe how AI or signal systems use wave mathematics.

Explain why trig graphs matter in engineering.

Explain why sound and light can be modeled mathematically.

Explain why sine and cosine graphs became foundational in physics and technology.

10.27 Solutions

Solutions to Warm-Up Problems

A function that repeats in cycles.

Wave height from midline.

Length of one full cycle.

Center line of oscillation.

1

2π

4

y = 0

Rotation repeats cyclically.

Circular motion creates repeating coordinate behavior.

Rotational geometry naturally produces oscillation.

Many natural and technological systems repeat cyclically.

Solutions to Guided Problems

5

3

2π

2π

0

1

1

-1

Air pressure oscillates periodically.

Alternating current oscillates repeatedly over time.

Examples include:

tides

seasons

sound

rotating systems

Trig graphs modeled waves, motion, and oscillatory systems mathematically.

Solutions to Challenge Problems

Key points:

amplitude = 2

period = 2π

Key points:

amplitude = 3

period = 2π

amplitude = 7

period = 2π

amplitude = 2

period = 2π

Rotating coordinates oscillate repeatedly, producing wave patterns.

Nature contains many cyclic and oscillatory systems.

AI and communications systems process wave-based signals mathematically.

Engineering constantly analyzes vibration, rotation, and periodic systems.

Sound and light exhibit repeating oscillatory behavior.

Sine and cosine graphs became foundational because waves, oscillation, and periodic behavior appear throughout physics, engineering, communications, and modern technology.