3.4-3.5 Sine and Cosine Graphs and Sinusoidal Functions

Utkarsh
February 3, 2025

Introduction

Sine and cosine graphs is essential for understanding periodic functions, a key concept in trigonometry and real-world applications. These functions describe cyclic behavior, such as the movement of pendulums, the alternating current in electricity, and even sound waves.

In this blog, we will explore:
✔️ The unit circle and how it helps define sine and cosine values.
✔️ Step-by-step graphing of sine and cosine functions.
✔️ Important characteristics like period, amplitude, midline, and frequency.
✔️ Sinusoidal transformations and how they shift, stretch, and compress the graphs.

By the end of this post, you will have a clear understanding of sine and cosine functions, which is an integral part of everything you need to know about AP Precalculus.

1. Step-by-Step Graphing of Sine and Cosine Functions

Now that we understand the unit circle values in previous blogs, we can use them to plot the sine and cosine graphs step by step. The graphs of these functions repeat periodically and exhibit distinct characteristics such as amplitude, period, midline, and frequency.

Step 1: Understanding the General Forms

The standard forms of the sine and cosine functions are:

$y = A \sin(Bx + C) + D$

$y = A \cos(Bx + C) + D$

where:

$A$ (Amplitude): Determines the height of the wave from the midline.
$B$ (Frequency): Affects the period of the function. The period is given by $\frac{2\pi}{B}$ .
$C$ (Phase Shift): Moves the graph left or right.
$D$ (Midline Shift): Moves the graph up or down.

For now, let’s start with the basic sine and cosine graphs without transformations.

Step 2: Constructing a Value Table

$x (radians)$	$\sin x$	$\cos x$
$0$	$0$	$1$
$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$
$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
$\frac{\pi}{2}$	$1$	$0$
$\pi$	$0$	$-1$
$\frac{3\pi}{2}$	$-1$	$0$
$2\pi$	$0$	$1$

Step 3: Plotting $y=\sin x$

The sine function starts at $(0,0)$ , reaches its peak at $(\frac{\pi}{2},1)$ , returns to zero at $(\pi,0)$ , dips to its minimum at $(\frac{3\pi}{2},-1)$ , and completes the cycle at $(2\pi,0)$ .
The function is periodic with a period of $2\pi$ and an amplitude of 1.

Step 4: Plotting $y = \cos x$

The cosine function starts at (0,1), reaches zero at $(\frac{\pi}{2},0)$ , goes to $-1$ at $(\pi,-1)$ , returns to zero at $(\frac{3\pi}{2},0)$ , and completes the cycle at $(2\pi,1)$ .
Like sine, cosine has a period of $2\pi$ and amplitude $1$ .

Step 5: Observing Key Properties

Both graphs share the following key properties:

Amplitude: The maximum displacement from the midline is 1.
Period: The function repeats every $2\pi$ .
Symmetry:
- The sine function is odd: $\sin(-x) = -\sin x$ .
- The cosine function is even: $\cos(-x) = \cos x$ .

5. Period, Amplitude, Midline, and Frequency

When analyzing the graphs of sine and cosine functions, we must consider four important properties:

Amplitude – The maximum height of the wave from the midline.
Period – The distance required for the function to complete one full cycle.
Midline – The horizontal axis around which the function oscillates.
Frequency – The number of cycles the function completes in a given interval.

Each of these properties is influenced by coefficients in the equations of $y = A \sin(Bx + C) + D$ and $y = A \cos(Bx + C) + D$ .

1. Amplitude ( $A$ )

The amplitude of a sine or cosine function is the absolute value of $A$ in the general equation:

$y = A \sin(Bx + C) + D$

$y = A \cos(Bx + C) + D$

Formula: $\text{Amplitude} = |A|$
Effect on Graph:
- A larger $A$ results in a taller wave.
- A smaller $A$ results in a shorter wave.
- If $A$ is negative, the graph is reflected over the midline.
Example:
- If $A = 2$ , the graph stretches twice as tall.
- If $A = \frac{1}{2}$ , the wave is half as tall.

The blue graph represents $A = 2$ , and the red graph represents $A = \frac{1}{2}$ .

2. Period ( $T$ )

The period is the length of one complete cycle of the wave. It depends on the value of $B$ :

$\text{Period} = \frac{2\pi}{B}$

Effect on Graph:
- A larger $B$ results in a shorter period, meaning more cycles occur in the same interval.
- A smaller $B$ results in a longer period, meaning fewer cycles occur.
Example:
- If $B = 1$ , the function has a period of $2\pi$ .
- If $B = 2$ , the function completes two cycles in $2\pi$ (shorter period).
- If $B = \frac{1}{2}$ , the function takes twice as long to complete one cycle.

The blue graph represents $B = 1$ (normal period), while the red graph represents $B = 2$ , meaning it completes a cycle twice as fast.

3. Midline ( $D$ )

The midline is the horizontal line around which the sine and cosine functions oscillate. It is determined by the value of $D$ in the equation:

$y = A \sin(Bx + C) + D$

Formula: $\text{Midline} = y = D$
Effect on Graph:
- A higher $D$ shifts the graph up.
- A lower $D$ shifts the graph down.
Example:
- If $D = 1$ , the midline moves up to $y = 1$ .
- If $D = -2$ , the midline shifts down to $y = -2$ .

4. Frequency ( $B$ )

The frequency tells us how many complete cycles occur in a given interval. It is given by:

$\text{Frequency} = \frac{B}{2\pi}$

Effect on Graph:
- A higher frequency $(B > 1)$ means more cycles occur.
- A lower frequency $(0 < B < 1)$ means fewer cycles occur.
Example:
- If $B = 1$ , the function completes one full cycle in $2\pi$ .
- If $B = 3$ , it completes three full cycles in $2\pi$ .
- If $B = \frac{1}{2}$ , it takes twice as long to complete a cycle.

Summary Table of Effects

Property	Formula	Effect on Graph
Amplitude (A)	$\|A\|$	Determines the vertical stretch or compression. If ∣A∣>1, the wave is taller (stretched). If 0<∣A∣<1, the wave is shorter (compressed). If A<0, the graph reflects across the x-axis.
Period (T)	$\frac{2\pi}{B}$	Determines the width of one cycle. If ∣B∣>1, the wave repeats more frequently (shorter period). If 0<∣B∣<1, the wave stretches out (longer period).
Midline (D)	$y=D$	Shifts the entire wave up or down. If D>0, the graph moves up. If D<0, the graph moves down. The midline is the horizontal axis around which the wave oscillates.
Frequency (B)	$B$	Determines how many cycles occur in a given interval. Larger B means more cycles (faster oscillation). Smaller B means fewer cycles (slower oscillation). Sometimes written as 2πB if measured in cycles per unit length.

6. Sinusoidal Functions and Transformations

Sinusoidal functions are a broad class of functions that describe periodic behavior. The most common sinusoidal functions are sine and cosine, but their transformations allow us to model real-world phenomena like sound waves, tides, electrical currents, and even seasonal temperature variations.

A transformed sinusoidal function is given by the general equation:

$y = A \sin(Bx + C) + D$

$y = A \cos(Bx + C) + D$

where:

$A$ (Amplitude) controls the height of the wave.
$B$ (Frequency/Period) determines the number of cycles in a given interval.
$C$ (Phase Shift) moves the wave left or right.
$D$ (Midline Shift) moves the wave up or down.

Let’s break down how these transformations affect the graph.

1. Amplitude Transformation ( $A$ )

The amplitude controls the vertical stretch or compression of the sine and cosine functions.

If $|A| > 1$ , the graph is stretched vertically.
If $0 < |A| < 1$ , the graph is compressed.
If $A < 0$ , the graph is reflected across the midline.

Example:

$y = 2\sin(x)$ → stretched (twice as tall)
$y = 0.5\cos(x)$ → compressed (half as tall)
$y = -\sin(x)$ → reflected over the x-axis

2. Period and Frequency Transformation ( $B$ )

The period of the function determines how long it takes for the wave to complete one full cycle. It is given by:

$\text{Period} = \frac{2\pi}{|B|}$

If $B > 1$ , the wave shrinks, completing more cycles in the same space.
If $0 < B < 1$ , the wave stretches, taking longer to complete a cycle.

Example:

$y = \sin(2x)$ → Half the normal period (more cycles).
$y = \cos\left(\frac{1}{2}x\right)$ → Twice the normal period (fewer cycles).

Blue: Standard sine wave.
Red: Higher frequency $(B = 2)$ → More cycles.
Green: Lower frequency $(B = 0.5)$ → Fewer cycles.

3. Phase Shift ( $C$ ) – Horizontal Shift

The phase shift moves the wave left or right based on the value of $C$ in the equation:

$y = A \sin(Bx + C) + D$

Formula for Phase Shift: $\text{Phase Shift} = \frac{-C}{B}$
If $C > 0$ , the graph shifts left.
If $C < 0$ , the graph shifts right.

Example:

$y = \sin(x - \frac{\pi}{3})$
$y = \cos(x + \frac{\pi}{2})$

4. Vertical Shift ( $D$ ) – Moving the Midline

The midline is shifted up or down depending on $D$ .

If $D > 0$ , the graph moves up.
If $D < 0$ , the graph moves down.

Example:

$y = \sin(x) + 2$ → Midline at $y = 2$ (shifted up).
$y = \cos(x) - 1$ → Midline at $y = -1$ (shifted down).

5. Combined Transformations

When all transformations are applied together, we get:

$y = A \sin(Bx + C) + D$

Transformation Effects

Amplitude ( $A$ ): Determines height of the wave.
Period ( $B$ ): Determines width of one cycle.
Phase Shift ( $C$ ): Moves the wave left or right.
Vertical Shift ( $D$ ): Moves the midline up or down.

Example Graph of a Transformed Sine Function

Consider the function:

$y = 2 \sin(3x - \frac{\pi}{4}) + 1$

Amplitude: $|A| = 2$ → Wave is twice as tall.
Period: $\frac{2\pi}{B} = \frac{2\pi}{3}$ → Completes one cycle faster.
Phase Shift: $-\frac{C}{B} = \frac{\pi}{12}$ → Shifted right by $\frac{\pi}{12}$ .
Midline: $D = 1$ → Shifted up.

This function represents a stretched, shifted, and faster sine wave.

Frequently Asked Questions (FAQs)

Que-1. Is there any free resource to self learn AP Precalculus?

Yes, you can download topic wise worksheets from our website.

Que-2. Do you offer AP Precalculus tutoring?

Yes, you can connect with an expert AP Precalculus private tutor for one on one tutoring. Feel free to contact here .

Keep an eye on the official College Board web page for current information regarding the Exam!

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3.4-3.5 Sine and Cosine Graphs and Sinusoidal Functions

Introduction