The Cosine Function Revision Notes for Grade 11 NSC Matric Mathematics

The Cosine Function

Introduction to the cosine function

The cosine function is one of the fundamental trigonometric functions that creates a smooth, repeating wave pattern. Understanding how to work with cosine functions and their transformations is essential for success in NSC Mathematics.

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The basic cosine function shows a characteristic wave shape that starts at its maximum value, decreases to its minimum, then returns to maximum, creating one complete cycle. This distinguishes it from the sine function, which starts at zero.

Basic cosine function: y = cos θ

Key properties of y = cos θ

The standard cosine function for the domain $0° ≤ θ ≤ 360°$ has several important characteristics that you must memorise:

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Essential Properties of $y = \cos θ$ :

Period: $360°$ (one complete wave cycle)
Amplitude: $1$ (maximum distance from the centre line)
Domain: All real numbers ( $θ ∈ ℝ$ ), but often restricted to $[0°; 360°]$ in problems
Range: $[-1; 1]$ (the function output values are between $-1$ and $1$ )

Important points on the basic cosine graph

Understanding the key points on the cosine graph helps you sketch it accurately and solve problems efficiently:

y-intercept: $(0°; 1)$ - the cosine function starts at its maximum value
x-intercepts: $(90°; 0)$ and $(270°; 0)$ - where the curve crosses the horizontal axis
Maximum turning points: $(0°; 1)$ and $(360°; 1)$ - the highest points on the curve
Minimum turning point: $(180°; -1)$ - the lowest point on the curve

Vertical transformations: y = a cos θ + q

The general form $y = a \cos θ + q$ involves two parameters that transform the basic cosine function vertically.

Effect of parameter q (vertical shift)

The parameter q moves the entire graph up or down:

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Vertical Shift Effects:

When $q > 0$ : the graph shifts upward by $q$ units
When $q < 0$ : the graph shifts downward by $|q|$ units
The range becomes $[q - 1; q + 1]$

Effect of parameter a (amplitude and reflection)

The parameter a affects both the height and orientation of the wave:

For $a > 1$ : The amplitude increases, making the wave taller

For $0 < a < 1$ : The amplitude decreases, making the wave shorter

For $a < 0$ : The graph reflects about the x-axis (flips upside down)

For $-1 < a < 0$ : Reflection occurs with decreased amplitude
For $a < -1$ : Reflection occurs with increased amplitude

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The amplitude of the transformed function is $|a|$ , and the range becomes $[q - |a|; q + |a|]$ .

Period transformations: y = cos(kθ)

When we write the cosine function as $y = \cos(kθ)$ , the parameter k affects how often the wave pattern repeats.

Effects of parameter k on the period

For $k > 0$ :

When $k > 1$ : the period decreases (wave cycles more frequently)
When $0 < k < 1$ : the period increases (wave cycles less frequently)

For $k < 0$ :

The period still changes according to $|k|$
However, $\cos(-θ) = \cos θ$ , so negative angles don't create reflections

Period formula

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Essential Formula:

The period of $y = \cos(kθ)$ is calculated using:

$\text{Period} = \frac{360°}{|k|}$

This formula is essential for exam questions and must be memorised.

Worked example: comparing y₁ = cos θ and y₂ = cos(θ/2)

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Worked Example: Comparing Cosine Functions

Let's compare $y₁ = \cos θ$ and $y₂ = \cos(θ/2)$ systematically:

Step 1: Identify the parameter values

For $y₁ = \cos θ$ : $k = 1$
For $y₂ = \cos(θ/2)$ : $k = 1/2$

Step 2: Calculate the periods

Period of $y₁ = 360°/|1| = 360°$
Period of $y₂ = 360°/|1/2| = 360°/(1/2) = 720°$

Step 3: Compare the characteristics

Property	$y₁ = \cos θ$	$y₂ = \cos(θ/2)$
Period	$360°$	$720°$
Amplitude	$1$	$1$
Domain	$[-180°; 180°]$	$[-180°; 180°]$
Range	$[-1; 1]$	$[0; 1]$
Maximum turning points	$(0°; 1)$	$(0°; 1)$
Minimum turning points	$(-180°; -1), (180°; -1)$	none in given domain

Phase shift transformations: y = cos(θ + p)

The form $y = \cos(θ + p)$ creates a horizontal shift (also called a phase shift) of the entire cosine graph.

Effect of parameter p

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Phase Shift Rules:

For $p > 0$ : The graph shifts to the left by $p$ degrees
For $p < 0$ : The graph shifts to the right by $|p|$ degrees

This horizontal movement affects the positions of all key points (intercepts, turning points) but doesn't change the period, amplitude, domain, or range.

Worked example: comparing y₁ = cos θ and y₂ = cos(θ + 30°)

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Worked Example: Phase Shift Comparison

Step 1: Identify the transformation

$y₁ = \cos θ$ has no phase shift ( $p = 0$ )
$y₂ = \cos(θ + 30°)$ has $p = 30°$ , so shifts left by $30°$

Step 2: Create a table of values

θ	$-360°$	$-270°$	$-180°$	$-90°$	$0°$	$90°$	$180°$	$270°$	$360°$
$\cos θ$	$1$	$0$	$-1$	$0$	$1$	$0$	$-1$	$0$	$1$
$\cos(θ + 30°)$	$0.87$	$-0.5$	$-0.87$	$0.5$	$0.87$	$-0.5$	$-0.87$	$0.5$	$0.87$

Step 3: Compare the graphs

The phase-shifted function maintains the same wave shape but with all points moved $30°$ to the left.

Worked example: sketching a complex cosine function

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Worked Example: Sketching Complex Transformations

Let's work through sketching $f(θ) = \cos 3(θ - 60°)$ step by step.

Question: Sketch the graph of $f(θ) = \cos 3(θ - 60°)$ for $0° ≤ θ ≤ 360°$ .

Step 1: Rewrite in standard form

$f(θ) = \cos 3(θ - 60°) = \cos(3θ - 180°)$

This shows us:

$k = 3$ (affects period)
The function is shifted right by $60°$

Step 2: Calculate key properties

Period: $360°/|3| = 120°$
Amplitude: $1$
Domain: $[0°; 360°]$
Range: $[-1; 1]$

Step 3: Create a table of values

θ	$0°$	$45°$	$90°$	$135°$	$180°$	$225°$	$270°$	$315°$	$360°$
$f(θ)$	$-1$	$0.71$	$0$	$-0.71$	$1$	$-0.71$	$0$	$0.71$	$-1$

Step 4: Plot and sketch

The graph shows three complete cycles within $360°$ due to the factor of $3$ , with the characteristic phase shift.

Finding equations from cosine graphs

Sometimes you'll need to work backwards from a graph to find the equation. Here's a systematic approach that will help you solve these problems efficiently:

Worked example: determining equation parameters

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Worked Example: Graph to Equation

Given a cosine graph, determine the values of $a$ , $k$ , and $p$ for $y = a \cos(kθ + p)$ .

Step 1: Find the period to determine $k$ Count the horizontal distance for one complete cycle. If the period is $360°$ , then $k = 1$ .

Step 2: Find the amplitude to determine $a$
Measure the vertical distance from the centre line to the maximum. If this distance is $2$ , then $a = 2$ .

Step 3: Find the phase shift to determine $p$ Compare with the standard cosine function. If the maximum occurs at $45°$ instead of $0°$ , the graph has shifted right by $45°$ , so $p = -45°$ .

Step 4: Write the final equation Combining all parameters: $y = 2 \cos(θ - 45°)$

Key formulas and exam tips

Essential formulas to memorise

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Critical Formulas for Exams:

Period formula: Period $= \frac{360°}{|k|}$
Basic cosine: $y = \cos θ$ has period $360°$ , amplitude $1$
General form: $y = a \cos(k(θ + p)) + q$
Even function property: $\cos(-θ) = \cos θ$

Exam strategies

When working with cosine functions in exams, follow these systematic approaches:

When sketching graphs:

Always start by identifying the period using $360°/|k|$
Mark the amplitude as $|a|$ units from the centre line
Apply phase shifts by moving key points horizontally
Apply vertical shifts by moving the entire graph up or down

When finding equations:

Determine period first, then calculate $k$
Find amplitude from the graph's height
Identify shifts by comparing with standard positions
Check your answer by substituting key points

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Common Exam Traps to Avoid:

Forgetting absolute value bars in $|k|$ for period calculations
Confusing left/right shifts with positive/negative $p$ values
Not adjusting the range when amplitude or vertical shifts change
Mixing up cosine (starts at maximum) with sine (starts at zero)

Remember!

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Key Points to Remember:

The basic cosine function $y = \cos θ$ has period $360°$ , amplitude $1$ , and starts at its maximum value.
The period of $y = \cos(kθ)$ is always $\frac{360°}{|k|}$ , where $|k|$ means the absolute value of $k$ .
Parameter transformations: ' $a$ ' changes amplitude, ' $q$ ' shifts vertically, ' $k$ ' changes period, and ' $p$ ' creates phase shifts.
Cosine is an even function, meaning $\cos(-θ) = \cos θ$ , so the graph is symmetric about the y-axis.
Always check your domain and range carefully when transformations are applied, as they often change these fundamental properties.

The Cosine Function (Grade 11 NSC Matric Mathematics): Revision Notes