Lesson 7.4: Graphs of Trigonometric Functions

Introduction

Trigonometry is a vital branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right-angled triangles. In this lesson, we will explore the graphs of the primary trigonometric functions: sine, cosine, and tangent. Understanding how these graphs behave, including their periods, amplitudes, and transformations, is crucial for modeling periodic phenomena in science and engineering.

Learning Objectives

By the end of this lesson, students will be able to:

Describe the graphs of sine, cosine, and tangent, including their periods and amplitudes.
Perform simple transformations of trigonometric graphs.
Model periodic behavior at an introductory level.
Sketch the graphs of sine, cosine, and tangent and state their key features.
Apply simple transformations to trigonometric graphs.

Section 1: Graphs of Sine and Cosine

1.1 The Sine Function

The sine function is defined in relation to the unit circle and right triangles. For a given angle $\theta$, the sine of $\theta$ corresponds to the ratio of the length of the opposite side to the hypotenuse in a right triangle.

1.2 Key Features of the Sine Graph

Shape: The graph is a smooth, continuous wave.
Period: The period of the sine function is $2\pi$. This means the graph repeats every $2\pi$ units along the x-axis.
Amplitude: The amplitude is 1, which indicates the maximum height of the graph is 1 and the minimum height is -1.

1.3 Graph of the Sine Function

The graph of the sine function can be plotted by evaluating $\sin(\theta)$ at various angles. Here’s a table of values:

Angle ($\theta$)	Sine ($\sin(\theta)$)
$0$	$0$
$\frac{\pi}{2}$	$1$
$\pi$	$0$
$\frac{3\pi}{2}$	$-1$
$2\pi$	$0$

We can plot these points on a graph:

$$\text{Graph of } y = \sin(x)$$

1.4 Worked Example: Graphing the Sine Function

Let's sketch the sine function over one period:

Start with the x-axis and y-axis.
Mark the key angles on the x-axis: $0$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and $2\pi$.
Plot the corresponding sine values on the y-axis.
Connect the points smoothly, creating a wave-like shape.

1.5 The Cosine Function

Similar to sine, the cosine function is also derived from the unit circle. For a given angle $\theta$, cosine represents the ratio of the length of the adjacent side to the hypotenuse.

1.6 Key Features of the Cosine Graph

Shape: The graph is also a smooth, continuous wave.
Period: The period of the cosine function is also $2\pi$.
Amplitude: Like sine, the amplitude of the cosine function is 1.

1.7 Graph of the Cosine Function

Using similar values as before, here are the key points for the cosine function:

Angle ($\theta$)	Cosine ($\cos(\theta)$)
$0$	$1$
$\frac{\pi}{2}$	$0$
$\pi$	$-1$
$\frac{3\pi}{2}$	$0$
$2\pi$	$1$

Now we can plot these points on a graph:

$$\text{Graph of } y = \cos(x)$$

1.8 Worked Example: Graphing the Cosine Function

To graph the cosine function:

Begin with the axes.
Mark key angles on the x-axis.
Plot the corresponding cosine values on the y-axis.
Smoothly connect the points, producing the classic cosine wave.

Section 2: The Tangent Function

2.1 Definition of the Tangent Function

The tangent function is defined as the ratio of sine to cosine: $ an(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $.

2.2 Key Features of the Tangent Graph

Shape: The graph has a repeating pattern of vertical asymptotes.
Period: The period of the tangent function is $\pi$.
Amplitude: Tangent does not have a fixed amplitude as its values range from $-\infty$ to $+\infty$.

2.3 Graph of the Tangent Function

We can observe the behavior of the tangent function:

Angle ($\theta$)	Tangent ($ an(\theta)$)
$0$	$0$
$\frac{\pi}{4}$	$1$
$\frac{\pi}{2}$	Undefined (asymptote)
$\frac{3\pi}{4}$	$-1$
$\pi$	$0$

As a result, we can sketch the graph of the tangent function considering the asymptotes at $\frac{\pi}{2}$, $\frac{3\pi}{2}$, etc.:

$$\text{Graph of } y = an(x)$$

2.4 Worked Example: Graphing the Tangent Function

To sketch the tangent function:

Draw the x and y axes.
Identify key angles and plot the tangent values, noting the asymptotes.
Connect the points, illustrating the steep rise and fall near the asymptotes.

Section 3: Simple Transformations of Trigonometric Graphs

3.1 Vertical Transformations

Vertical transformations involve shifting the graph up or down. The function can be modified by adding or subtracting a constant:

$$y = \sin(x) + k$$

If $k > 0$, the graph shifts $k$ units up.
If $k < 0$, the graph shifts $|k|$ units down.

3.2 Horizontal Transformations

Horizontal transformations involve shifting the graph left or right and are represented as follows:

$$y = \sin(x - h)$$

If $h > 0$, the graph shifts $h$ units to the right.
If $h < 0$, the graph shifts $|h|$ units to the left.

3.3 Amplitude and Period Changes

The amplitude and period can also be altered by multiplying the sine function by a coefficient $a$ and adjusting the input:

$$y = a\sin(bx)$$

The amplitude is $|a|$; if $a > 0$, the wave peaks at $a$.
The period changes to $\frac{2\pi}{|b|}$; higher $|b|$ compresses the graph.

3.4 Worked Example: Transforming the Sine Function

Let’s transform $y = \sin(x)$ into $y = 2\sin(x - \frac{\pi}{2}) + 1$.

Shift the graph right by $\frac{\pi}{2}$.
Stretch the graph vertically by a factor of 2.
Shift the entire graph up by 1.

Conclusion

In this lesson, we explored the graphs of the sine, cosine, and tangent functions, focusing on their periodic nature, amplitude, and fundamental transformations. Understanding these primary trigonometric functions sets students up for success in more advanced topics within trigonometry and its applications.

Study Notes

The sine graph oscillates between -1 and 1, with a period of $2\pi$.
The cosine graph similarly oscillates between -1 and 1, also with a period of $2\pi$.
The tangent graph fluctuates between $-\infty$ and $+\infty$, with vertical asymptotes at $(\frac{\pi}{2} + k\pi)$ where $k$ is any integer.
Transformations can involve shifting, stretching, or compressing the graphs as needed.