Graphs of Trigonometric Functions

Trigonometric functions produce some of the most recognisable graphs in mathematics. Unlike linear or quadratic functions, trigonometric functions often repeat their values in regular patterns.

These repeating patterns make trigonometric graphs especially useful for describing phenomena such as sound waves, ocean tides, alternating electrical currents, rotating wheels, seasonal temperatures, and mechanical vibrations.

The three principal trigonometric graphs are:

• Sine: y = sin x

• Cosine: y = cos x

• Tangent: y = tan x

Sine and cosine produce smooth, continuous waves. Tangent produces repeating branches separated by vertical asymptotes.

Understanding these graphs involves identifying their key points, amplitude, period, midline, intercepts, and transformations.

The Sine Graph

The sine function is:

y = sin x

The sine of an angle corresponds to the y-coordinate of a point moving around the unit circle.

As the point completes one revolution, its vertical coordinate changes from 0 to 1, back to 0, down to −1, and finally back to 0.

This produces the familiar sine wave.

Key Points of the Sine Graph

These five points describe one complete cycle.

The graph:

1. Begins at the midline.

2. Rises to a maximum.

3. Returns to the midline.

4. Falls to a minimum.

5. Returns to the midline.

The pattern repeats indefinitely in both directions.

Properties of y = sin x

• Domain: All real numbers

• Range: −1 ≤ y ≤ 1

• Amplitude: 1

• Period: 2π radians or 360°

• Midline: y = 0

• Maximum value: 1

• Minimum value: −1

The sine graph is continuous, meaning it has no breaks or jumps.

The Cosine Graph

The cosine function is:

y = cos x

Cosine represents the x-coordinate of a point moving around the unit circle.

At an angle of 0°, the point is at (1, 0), so cosine begins at its maximum value of 1.

Key Points of the Cosine Graph

One complete cosine cycle follows this pattern:

1. Begins at a maximum.

2. Decreases to the midline.

3. Reaches a minimum.

4. Increases to the midline.

5. Returns to the maximum.

Properties of y = cos x

• Domain: All real numbers

• Range: −1 ≤ y ≤ 1

• Amplitude: 1

• Period: 2π radians or 360°

• Midline: y = 0

• Maximum value: 1

• Minimum value: −1

Sine and cosine have the same shape, amplitude, period, and range.

Their main difference is their horizontal position.

Comparing Sine and Cosine

Sine and cosine are closely related.

In fact:

cos x = sin(x + π/2)

This means the cosine graph can be obtained by shifting the sine graph π/2 radians to the left.

Similarly:

sin x = cos(x − π/2)

The sine graph can be obtained by shifting the cosine graph π/2 radians to the right.

Both functions describe periodic behaviour and can model the same types of phenomena using different phase shifts.

The Tangent Graph

The tangent function is:

y = tan x

Tangent is defined by:

tan x = sin x/cos x

Unlike sine and cosine, tangent does not have a maximum or minimum value.

Its values can become arbitrarily large in the positive or negative direction.

Key Points of the Tangent Graph

The graph increases continuously between consecutive vertical asymptotes.

Vertical Asymptotes

Tangent is undefined when cosine equals zero.

This occurs at:

x = π/2 + nπ

where n is any integer.

Examples include:

−3π/2, −π/2, π/2, 3π/2

These values correspond to vertical asymptotes.

Properties of y = tan x

• Domain: All real numbers except x = π/2 + nπ

• Range: All real numbers

• Period: π radians or 180°

• Midline: y = 0

• Zeros: x = nπ

• Vertical asymptotes: x = π/2 + nπ

• Amplitude: Not defined

Tangent has no amplitude because it is unbounded.

Understanding Amplitude

The amplitude of a sine or cosine graph is the maximum vertical distance between the midline and a maximum or minimum point.

For:

y = A sin x

or:

y = A cos x

the amplitude is:

Amplitude = |A|

Worked Example

Consider:

y = 3 sin x

The coefficient is:

A = 3

Therefore:

Amplitude = 3

The maximum value is 3, and the minimum value is −3.

Worked Example

Consider:

y = −4 cos x

The amplitude is:

|−4| = 4

Therefore:

Amplitude = 4

The negative sign reflects the graph across its midline but does not make the amplitude negative.

Amplitude from Maximum and Minimum Values

If a graph has maximum value M and minimum value m:

Amplitude = (M − m)/2

For example, suppose a wave reaches a maximum of 9 and a minimum of 1.

Amplitude:

(9 − 1)/2 = 4

Amplitude = 4

Understanding the Midline

The midline is the horizontal line halfway between the maximum and minimum values of a sinusoidal graph.

For:

y = A sin(Bx + C) + D

or:

y = A cos(Bx + C) + D

the midline is:

y = D

The midline represents the central or equilibrium value around which the graph oscillates.

Worked Example

Consider:

y = 2 sin x + 5

The midline is:

y = 5

The amplitude is 2.

Therefore:

Maximum = 5 + 2 = 7

Minimum = 5 − 2 = 3

The graph oscillates between 3 and 7.

Midline from Maximum and Minimum

Midline value = (Maximum + Minimum)/2

If the maximum is 12 and the minimum is 4:

Midline = (12 + 4)/2

Midline: y = 8

The amplitude is:

(12 − 4)/2 = 4

Understanding Period

The period is the horizontal distance required for a periodic function to complete one full cycle.

The standard sine and cosine functions have a period of:

2π radians or 360°

The standard tangent function has a period of:

π radians or 180°

Sine and Cosine

For:

y = A sin(Bx + C) + D

or:

y = A cos(Bx + C) + D

the period is:

Period = 2π/|B|

when x is measured in radians and B ≠ 0.

In degrees:

Period = 360°/|B|

Tangent

For:

y = A tan(Bx + C) + D

the period is:

Period = π/|B|

In degrees:

Period = 180°/|B|

Worked Example

Find the period of:

y = sin(2x)

Here:

B = 2

Period = 2π/2

Period = π

The graph completes two full cycles between 0 and 2π.

Worked Example

Find the period of:

y = cos(x/3)

Here:

B = 1/3

Period = 2π/(1/3)

Period = 6π

The graph is stretched horizontally.

Worked Example

Find the period of:

y = tan(4x)

Period = π/4

Period = π/4

The tangent graph repeats four times as frequently as the standard tangent graph.

Transformations of Sine and Cosine Graphs

A general sinusoidal function can be written as:

y = A sin[B(x − h)] + D

or:

y = A cos[B(x − h)] + D

where:

• A controls amplitude and vertical reflection.

• B controls period and horizontal scaling.

• h controls horizontal translation.

• D controls vertical translation.

Vertical Stretch and Compression

The coefficient A changes the amplitude.

If |A| > 1, the graph is stretched vertically.

If 0 < |A| < 1, the graph is compressed vertically.

For example:

y = 5 sin x

has amplitude 5.

Meanwhile:

y = 0.5 sin x

has amplitude 0.5.

Reflection

If A is negative, the graph is reflected across its midline relative to the corresponding positive-A graph.

For example:

y = −sin x

is the reflection of y = sin x across the x-axis.

Horizontal Scaling

The coefficient B changes the period.

If |B| > 1, the period becomes shorter.

If 0 < |B| < 1, the period becomes longer.

Horizontal Translation

The value h shifts the graph horizontally.

For example:

y = sin(x − π/3)

is shifted π/3 radians to the right.

Meanwhile:

y = cos(x + π/4)

is shifted π/4 radians to the left.

Vertical Translation

The value D shifts the graph vertically.

For example:

y = sin x + 3

has midline y = 3.

Worked Example: Analysing a Sinusoidal Function

Consider:

y = 3 sin[2(x − π/4)] + 1

Determine the amplitude, period, horizontal shift, midline, maximum, and minimum.

Amplitude

A = 3

Amplitude = 3

Period

B = 2

Period = 2π/2

Period = π

Horizontal Shift

h = π/4

Shift = π/4 to the right

Midline

D = 1

Midline: y = 1

Maximum and Minimum

Maximum = D + |A|

= 1 + 3

Maximum = 4

Minimum = D − |A|

= 1 − 3

Minimum = −2

Interpretation

The graph oscillates around y = 1.

It reaches a maximum of 4 and a minimum of −2.

It completes one full cycle every π radians and is shifted π/4 radians to the right.

Sketching Sine and Cosine Graphs

An efficient way to sketch a sinusoidal graph is to identify five key points over one complete cycle.

For:

y = A sin[B(x − h)] + D

with B > 0, use:

• Starting point: x = h

• Quarter-cycle: x = h + P/4

• Half-cycle: x = h + P/2

• Three-quarter-cycle: x = h + 3P/4

• Full cycle: x = h + P

where:

P = 2π/B

Standard Sine Pattern

For positive A, the five y-values are:

D, D + A, D, D − A, D

For negative A, the same formula still works, but the first movement is downward.

Standard Cosine Pattern

For positive A, the five y-values are:

D + A, D, D − A, D, D + A

These patterns can be used to sketch transformed sine and cosine graphs without a calculator.

Worked Example: Sketching a Sine Graph

Sketch one complete cycle of:

y = 2 sin x − 1

Identify the Properties

Amplitude = 2

Period = 2π

Midline: y = −1

Maximum = 1

Minimum = −3

Determine Key Points

Plot the five points and connect them with a smooth curve.

The graph begins at the midline, rises to 1, returns to the midline, falls to −3, and returns to the midline.

Worked Example: Sketching a Cosine Graph

Sketch one cycle of:

y = −3 cos x + 2

Identify the Properties

Amplitude = 3

Period = 2π

Midline: y = 2

Maximum = 5

Minimum = −1

Determine Key Points

Because the cosine coefficient is negative, the graph begins at a minimum rather than a maximum.

Sketching Tangent Graphs

Tangent graphs require a different approach because they contain vertical asymptotes.

For:

y = A tan[B(x − h)] + D

with B ≠ 0:

• Period = π/|B|

• Midline: y = D

• Central point: (h, D)

• Vertical asymptotes occur at x = h + (π/2 + nπ)/B

A tangent graph does not have an amplitude.

Worked Example

Sketch:

y = 2 tan x

The period is:

π

The midline is:

y = 0

Vertical asymptotes occur at:

x = −π/2 and x = π/2

for the central branch.

Key Points

Draw the central branch through these points, increasing from negative infinity toward positive infinity between the asymptotes.

Repeat the pattern every π radians.

Comparing Sine, Cosine, and Tangent

Sine and cosine are bounded, continuous waves.

Tangent is unbounded and has repeating vertical asymptotes.

All three functions are periodic.

Finding an Equation from a Graph

Sometimes a graph is given and we must determine a suitable trigonometric equation.

For a sinusoidal graph, identify:

• Maximum value.

• Minimum value.

• Amplitude.

• Midline.

• Period.

• Horizontal position of a key point.

Worked Example

A sinusoidal graph has:

• Maximum value = 7

• Minimum value = −1

• Period = 4π

• A maximum at x = 0

Find a possible equation.

Amplitude

A = (7 − (−1))/2

A = 4

Midline

D = (7 + (−1))/2

D = 3

Period Coefficient

P = 4π

For cosine:

P = 2π/|B|

Therefore:

4π = 2π/|B|

|B| = 1/2

Choose the Function

Since the graph has a maximum at x = 0, a positive cosine function is convenient.

Therefore:

y = 4 cos(x/2) + 3

This equation has the required amplitude, midline, period, and starting maximum.

Interpreting Trigonometric Graphs in Context

Trigonometric functions are particularly useful for modelling periodic real-world phenomena.

Ferris Wheel Motion

A Ferris wheel has radius 12 m, and its centre is 15 m above the ground.

A passenger begins at the lowest point.

The wheel completes one revolution every 40 seconds.

The passenger's height can be modelled by:

h(t) = 15 − 12 cos(πt/20)

where h is measured in metres and t in seconds.

Interpret the Model

Amplitude = 12 m

This represents the radius of the wheel.

Midline = 15 m

This represents the height of the wheel's centre.

Period = 40 s

This represents the time required for one revolution.

Maximum height:

15 + 12 = 27 m

Minimum height:

15 − 12 = 3 m

Find the Height After 10 Seconds

h(10) = 15 − 12 cos(π/2)

Since cos(π/2) = 0:

h(10) = 15 m

After ten seconds, the passenger is level with the centre of the wheel.

Modelling Ocean Tides

Ocean tides rise and fall in repeating patterns.

A simplified tide model might be:

h(t) = 2 + 1.5 cos(πt/6)

where:

• h = water height in metres

• t = time in hours

Interpret the Model

Amplitude = 1.5 m

Midline = 2 m

Maximum height = 3.5 m

Minimum height = 0.5 m

Period:

P = 2π/(π/6)

P = 12 hours

The model predicts one complete cycle every 12 hours.

Find the Height After 3 Hours

h(3) = 2 + 1.5 cos(π/2)

h(3) = 2

Water height = 2 m

This is a simplified model. Actual tides are influenced by multiple astronomical and geographic factors.

Modelling Seasonal Temperature

Average temperature often follows an approximately periodic annual pattern.

A simplified model might be:

T(t) = 20 + 10 cos[2π(t − 7)/12]

where:

• T = average temperature in °C

• t = month number

• t = 1 represents January

Interpret the Model

Amplitude = 10°C

Midline = 20°C

Maximum temperature = 30°C

Minimum temperature = 10°C

Period = 12 months

The horizontal shift places the maximum at t = 7, corresponding to July.

This type of model is useful for representing seasonal changes, although actual temperatures are affected by weather variability and local climate.

Applications in Physics and Engineering

Trigonometric graphs appear throughout science and engineering.

Simple Harmonic Motion

An oscillating object may be modelled by:

x(t) = A cos(ωt + φ)

where:

• A = amplitude

• ω = angular frequency

• φ = phase constant

The period is:

P = 2π/ω

when ω > 0.

Alternating Current

An alternating electrical current may be described by:

I(t) = I₀ sin(ωt)

The current repeatedly changes direction.

Sound Waves

Sound pressure variations can be approximated using sinusoidal functions for pure tones.

Amplitude relates to the strength of the pressure variation, while frequency determines the pitch of the sound.

Common Mistakes

Confusing amplitude and period

Amplitude measures vertical distance from the midline. Period measures horizontal length of one cycle.

Assuming amplitude can be negative

Amplitude is always nonnegative. A negative coefficient indicates a reflection.

Using the sine or cosine period formula for tangent

Sine and cosine use 2π/|B|. Tangent uses π/|B|.

Confusing the midline with the x-axis

The midline may be above or below y = 0.

Forgetting horizontal shifts

A phase shift changes the positions of key points.

Assuming tangent has an amplitude

Tangent is unbounded, so amplitude is not defined.

Connecting tangent branches across asymptotes

A tangent graph must not cross a vertical asymptote.

Plotting too few points

Five key points are useful for sketching one complete sine or cosine cycle.

Using incorrect calculator angle settings

Radians and degrees must be handled consistently.

Ignoring units in contextual problems

Periods may be measured in seconds, hours, or months, depending on the situation.

Assuming all real-world periodic patterns are perfectly sinusoidal

Sinusoidal functions are useful approximations, but real data may contain irregularities.

Key Terms

Trigonometric Function: A function relating angles to numerical values, often describing periodic behaviour.

Sine Function: A periodic function based on the vertical coordinate of the unit circle.

Cosine Function: A periodic function based on the horizontal coordinate of the unit circle.

Tangent Function: A periodic function defined as the ratio of sine to cosine.

Periodic Function: A function whose values repeat after a fixed interval.

Cycle: One complete repetition of a periodic pattern.

Period: The horizontal distance required for one complete cycle.

Amplitude: The maximum vertical distance from the midline of a sinusoidal graph.

Midline: The horizontal line halfway between the maximum and minimum values.

Maximum Value: The greatest output value of a function over a specified interval.

Minimum Value: The smallest output value of a function over a specified interval.

Range: The set of possible output values of a function.

Domain: The set of permitted input values of a function.

Vertical Translation: A shift of a graph upward or downward.

Horizontal Translation: A shift of a graph left or right.

Phase Shift: The horizontal displacement of a periodic graph.

Vertical Stretch: A transformation that increases distances from the midline.

Vertical Compression: A transformation that decreases distances from the midline.

Reflection: A transformation that reverses the graph across a line.

Horizontal Scaling: A transformation that changes the period of a graph.

Vertical Asymptote: A vertical line approached by a graph as function values increase or decrease without bound.

Intercept: A point where a graph crosses or touches a coordinate axis.

Key Point: A point used to identify the shape and position of a graph.

Angular Frequency: The rate of angular change, commonly represented by ω.

Simple Harmonic Motion: Oscillatory motion described by a sinusoidal function.

Sinusoidal Model: A mathematical model using sine or cosine to represent repeating behaviour.

Check Your Understanding

1. Sketch one complete cycle of y = sin x from 0 to 2π. Label the five key points.

2. Sketch one complete cycle of y = cos x from 0 to 2π. Label the maximum and minimum values.

3. Explain why the tangent graph has vertical asymptotes.

4. State the amplitude, period, and midline of y = 4 sin x.

5. Determine the amplitude, period, and midline of y = 3 cos(2x) + 1.

6. Find the maximum and minimum values of y = −5 sin x + 2.

7. Determine the period of y = sin(3x).

8. Determine the period of y = tan(x/2).

9. Sketch one cycle of y = 2 sin x − 3 using five key points.

10. Sketch one cycle of y = −2 cos x + 1.

11. Determine the amplitude, period, phase shift, and midline of y = 4 sin[2(x − π/3)] + 2.

12. A sinusoidal graph has a maximum of 11 and a minimum of 3. Find its amplitude and midline.

13. A cosine graph has amplitude 5, period 6π, midline y = −2, and a maximum at x = 0. Write a possible equation.

14. A Ferris wheel has radius 10 m and its centre is 13 m above the ground. It completes one revolution every 30 seconds. Write a cosine model for a passenger starting at the lowest point.

15. A tide is modelled by:

h(t) = 3 + 2 sin(πt/6)

where h is measured in metres and t in hours.

Determine its amplitude, midline, period, maximum height, minimum height, and water height after 3 hours.

Key Takeaways

• Sine, cosine, and tangent are periodic trigonometric functions.

• The sine graph begins at zero and initially increases.

• The cosine graph begins at its maximum value of 1.

• Sine and cosine have the same wave shape but differ by a horizontal shift.

• Tangent has repeating branches separated by vertical asymptotes.

• The standard sine and cosine functions have amplitude 1 and period 2π.

• The standard tangent function has period π and no amplitude.

• Amplitude measures the maximum vertical distance from the midline.

• The midline is halfway between the maximum and minimum values.

• The period measures the horizontal length of one complete cycle.

• For y = A sin[B(x − h)] + D and y = A cos[B(x − h)] + D, amplitude is |A| and period is 2π/|B|.

• For y = A tan[B(x − h)] + D, period is π/|B|.

• The value h controls horizontal translation.

• The value D controls vertical translation and determines the midline.

• Negative values of A reflect sine and cosine graphs across their midlines.

• Five key points can be used to sketch one complete sine or cosine cycle.

• Tangent graphs require careful identification of vertical asymptotes.

• Trigonometric equations can be constructed from graph features such as amplitude, period, and midline.

• Sinusoidal functions can model Ferris wheels, tides, seasonal temperatures, waves, and oscillations.

• Real-world interpretations should include appropriate units and acknowledge model limitations.

• Understanding the graphical properties of trigonometric functions provides a foundation for trigonometric modelling, advanced algebra, physics, and calculus.