The Sine Function Revision Notes for Grade 11 NSC Matric Mathematics

The Sine Function

Introduction to the sine function

The sine function is one of the fundamental trigonometric functions that produces a smooth, wave-like curve. Understanding this function is essential for success in NSC Mathematics Paper 1.

Basic form: $y = \sin \theta$ where $\theta$ is measured in degrees.

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The sine function appears frequently in NSC examinations, particularly in questions involving sketching graphs, finding periods, and analyzing transformations. Mastering this function will give you a strong foundation for trigonometry.

Properties of the basic sine function

The standard sine function $y = \sin \theta$ has several key characteristics that you must memorise:

Period

The period is the length of one complete wave cycle. For the basic sine function, the period is 360°.

Amplitude

The amplitude is the maximum height of the wave above and below the x-axis. For $y = \sin \theta$ , the amplitude is 1.

Domain and range

Domain: $\{θ : θ ∈ ℝ\}$ (all real numbers)
Range: $[-1; 1]$ (all values between -1 and 1, inclusive)

Intercepts and turning points

y-intercept: $(0°; 0)$
x-intercepts: $(0°; 0)$ , $(180°; 0)$ , $(360°; 0)$ , and so on
Maximum turning point: $(90°; 1)$
Minimum turning point: $(270°; -1)$

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Critical Properties to Memorize:

Period = 360°
Amplitude = 1
Range = [-1; 1]
Key points: (0°; 0), (90°; 1), (180°; 0), (270°; -1), (360°; 0)

These form the foundation for understanding all sine function transformations.

Transformations of the sine function

Vertical transformations: $y = a \sin \theta + q$

The parameters $a$ and $q$ affect the sine function in specific ways:

Effect of parameter $a$ (amplitude changes)

When $a > 1$ : amplitude increases (wave gets taller)
When $0 < a < 1$ : amplitude decreases (wave gets shorter)
When $a < 0$ : graph reflects about the x-axis
When $-1 < a < 0$ : graph reflects and amplitude decreases
When $a < -1$ : graph reflects and amplitude increases

Effect of parameter $q$ (vertical shift)

When $q > 0$ : graph shifts upward by $q$ units
When $q < 0$ : graph shifts downward by $|q|$ units

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Understanding Parameters:

Parameter $a$ controls the height of the wave (amplitude and reflection)
Parameter $q$ controls the vertical position of the entire wave
These transformations work independently of each other

Period changes: $y = \sin k\theta$

The parameter $k$ affects the period of the sine function:

Formula for period

$\text{Period} = \frac{360°}{|k|}$

Effects of different values of $k$ :

When $k > 1$ : period decreases (more cycles in 360°)
When $0 < k < 1$ : period increases (fewer cycles in 360°)
When $k < 0$ : graph reflects about the y-axis

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Period Formula Rule: Always use the absolute value of $k$ when calculating the period. This is a common source of errors in examinations.

Phase shifts: $y = \sin(\theta + p)$

The parameter $p$ causes a horizontal shift or phase shift:

When $p > 0$ : graph shifts left by $p$ degrees
When $p < 0$ : graph shifts right by $|p|$ degrees

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Phase Shift Memory Tip: Think of phase shifts as the opposite of what you might expect:

Positive $p$ → moves LEFT
Negative $p$ → moves RIGHT

This is because we're adding/subtracting inside the function.

Worked example 1: Comparing sine functions

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Worked Example: Comparing Sine Functions

Question: Sketch $y₁ = \sin \theta$ and $y₂ = \sin \frac{3\theta}{2}$ for $-180° \leq \theta \leq 180°$ .

Solution:

Step 1: Identify the transformations

$y₁ = \sin \theta$ is the standard sine function
$y₂ = \sin \frac{3\theta}{2}$ has $k = \frac{3}{2}$ , so the period changes

Step 2: Calculate the new period

$\text{Period of } y₂ = \frac{360°}{|\frac{3}{2}|} = \frac{360°}{\frac{3}{2}} = 240°$

Step 3: Create a table of values

$\theta$	$-180°$	$-135°$	$-90°$	$-45°$	$0°$	$45°$	$90°$	$135°$	$180°$
$\sin \theta$	$0$	$-0.71$	$-1$	$-0.71$	$0$	$0.71$	$1$	$0.71$	$0$
$\sin \frac{3\theta}{2}$	$1$	$0.38$	$-0.71$	$-0.92$	$0$	$0.92$	$0.71$	$-0.38$	$-1$

Worked example 2: Phase shift

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Worked Example: Phase Shift Analysis

Question: Sketch $y₁ = \sin \theta$ and $y₂ = \sin(\theta - 30°)$ for $-360° \leq \theta \leq 360°$ .

Solution:

Step 1: Identify the transformation

$y₂ = \sin(\theta - 30°)$ has $p = -30°$
This means the graph shifts right by 30°

Step 2: Create a table of values

$\theta$	$-360°$	$-270°$	$-180°$	$-90°$	$0°$	$90°$	$180°$	$270°$	$360°$
$\sin \theta$	$0$	$1$	$0$	$-1$	$0$	$1$	$0$	$-1$	$0$
$\sin(\theta - 30°)$	$-0.5$	$0.87$	$0.5$	$-0.87$	$-0.5$	$0.87$	$0.5$	$-0.87$	$-0.5$

Worked example 3: Complex transformation

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Worked Example: Complex Transformation

Question: Sketch $f(\theta) = \sin 3(\theta + 20°)$ for $0° \leq \theta \leq 180°$ .

Solution:

Step 1: Rewrite in standard form

$f(\theta) = \sin 3(\theta + 20°) = \sin(3\theta + 60°)$

Step 2: Identify transformations

$k = 3$ , so period = $\frac{360°}{3} = 120°$
Phase shift left by 20°

Step 3: Calculate key values and sketch

The function completes 3 cycles in 360°, with each cycle taking 120°.

Sketching technique summary

When sketching sine functions, follow these systematic steps:

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Step-by-Step Sketching Method:

Identify the form: $y = a \sin k(\theta + p) + q$
Calculate the period: $\text{Period} = \frac{360°}{|k|}$
Determine transformations:
- Amplitude: $|a|$
- Vertical shift: $q$
- Phase shift: $p$
Create a table of values for key angles
Plot points and join with a smooth curve

Key formulas to remember

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Essential Formulas:

Period formula: $\text{Period} = \frac{360°}{|k|}$
Basic sine values: $\sin 0° = 0$ , $\sin 90° = 1$ , $\sin 180° = 0$ , $\sin 270° = -1$
Negative angles: $\sin(-\theta) = -\sin \theta$
Domain: All real numbers
Range: $[-|a| + q; |a| + q]$ for $y = a \sin k(\theta + p) + q$

Common exam traps

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Common Mistakes to Avoid:

Don't confuse period changes: Remember to use $|k|$ in the period formula
Phase shift direction: Positive $p$ moves the graph left, negative $p$ moves it right
Amplitude vs vertical shift: $a$ changes the height of the wave, $q$ moves the entire wave up or down
Always check your domain: Make sure your sketch covers the required interval

Remember!

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Key Points to Remember:

The period of $y = \sin k\theta$ is $\frac{360°}{|k|}$ degrees
Amplitude is always positive and equals $|a|$ for $y = a \sin \theta$
Phase shifts: $y = \sin(\theta + p)$ moves the graph left when $p > 0$ , right when $p < 0$
Vertical shifts: $y = \sin \theta + q$ moves the graph up when $q > 0$ , down when $q < 0$
Always create a table of values for key angles when sketching sine functions

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