Sketching Graphs of y = sin x, cos x, tan x

Introduction

Key Concepts

Basic Definitions and Properties

Trigonometric functions are periodic functions that relate the angles of a right triangle to the ratios of its sides. The primary functions—sine, cosine, and tangent—are defined as follows:

• Sine Function: For an angle $x$, $y = \sin x$ represents the y-coordinate of a point on the unit circle.
• Cosine Function: For an angle $x$, $y = \cos x$ represents the x-coordinate of a point on the unit circle.
• Tangent Function: For an angle $x$, $y = \tan x = \frac{\sin x}{\cos x}$, representing the slope of the terminal side of the angle.

These functions exhibit distinct periodicities and ranges:

• Sine and Cosine: Period of $2\pi$, range $[-1, 1]$.
• Tangent: Period of $\pi$, range $(-\infty, \infty)$.

Amplitude, Period, and Phase Shifts

Transformations of trigonometric functions involve changing their amplitude, period, and phase. The general form of these functions is:

$y = A \cdot \sin(Bx - C) + D$, $y = A \cdot \cos(Bx - C) + D$, and $y = A \cdot \tan(Bx - C) + D$,

where:

• A: Amplitude – controls the vertical stretch/compression.
• B: Affects the period – the new period is $\frac{2\pi}{B}$ for sine and cosine, and $\frac{\pi}{B}$ for tangent.
• C: Phase shift – horizontal shift of the graph.
• D: Vertical shift – moves the graph up or down.

Graphing y = sin x

The sine function is an odd function, symmetric about the origin. Its graph starts at $(0,0)$, reaches a maximum at $\frac{\pi}{2}$, crosses the x-axis at $\pi$, reaches a minimum at $\frac{3\pi}{2}$, and returns to $(2\pi,0)$. Key characteristics include:

• Amplitude: 1
• Period: $2\pi$
• Phase Shift: 0
• Vertical Shift: 0

To sketch $y = \sin x$, plot these key points and draw a smooth, continuous wave passing through them.

Graphing y = cos x

The cosine function is an even function, symmetric about the y-axis. Its graph starts at $(0,1)$, crosses the x-axis at $\frac{\pi}{2}$, reaches a minimum at $\pi$, crosses again at $\frac{3\pi}{2}$, and returns to $(2\pi,1)$. Key characteristics include:

• Amplitude: 1
• Period: $2\pi$
• Phase Shift: 0
• Vertical Shift: 0

To sketch $y = \cos x$, plot these key points and draw a smooth, continuous wave passing through them.

Graphing y = tan x

The tangent function has vertical asymptotes where $\cos x = 0$, specifically at $x = \frac{\pi}{2} + k\pi$ for any integer $k$. Its graph passes through the origin, increasing to positive infinity as it approaches the asymptote from the left and decreasing to negative infinity as it approaches from the right. Key characteristics include:

• Amplitude: Undefined (since range is all real numbers)
• Period: $\pi$
• Phase Shift: 0
• Vertical Shift: 0
• Asymptotes: $x = \frac{\pi}{2} + k\pi$

To sketch $y = \tan x$, draw the asymptotes and plot points between them to form the characteristic repeating pattern.

Transformations of Trigonometric Graphs

Transformations modify the basic trigonometric graphs by altering their amplitude, period, phase shift, and vertical shift. Applying transformations requires understanding the general form:

$y = A \cdot \sin(Bx - C) + D$, $y = A \cdot \cos(Bx - C) + D$, $y = A \cdot \tan(Bx - C) + D$.

• Amplitude Change: Multiplied by $A$, stretching or compressing vertically.
• Period Adjustment: Divided by $B$, altering the horizontal length of one cycle.
• Phase Shift: Shifted horizontally by $\frac{C}{B}$ units.
• Vertical Shift: Shifted vertically by $D$ units.

For example, $y = 2\sin(3x - \pi) + 1$ has:

• Amplitude: 2
• Period: $\frac{2\pi}{3}$
• Phase Shift: $\frac{\pi}{3}$ units to the right
• Vertical Shift: 1 unit upwards

Phase Shift and Frequency

Phase shifts move the graph left or right without altering its shape. A positive phase shift moves the graph to the right, while a negative shift moves it to the left. Frequency, related to the period, indicates how many cycles occur within $2\pi$ radians:

• Higher Frequency: More cycles within $2\pi$.
• Lower Frequency: Fewer cycles within $2\pi$.

Understanding these concepts allows for precise graphing of transformed trigonometric functions.

Amplitude and Vertical Stretching/Compression

Amplitude affects the height of the wave. Increasing the amplitude stretches the graph vertically, making peaks and troughs higher and lower, respectively. Decreasing the amplitude compresses the graph, making the wave less tall:

• $y = 3\sin x$ has an amplitude of 3.
• $y = 0.5\cos x$ has an amplitude of 0.5.

These transformations do not affect the period or phase of the function.

Vertical Shifts

Vertical shifts move the entire graph up or down by a constant value $D$:

• $y = \sin x + 2$ shifts the sine graph 2 units upwards.
• $y = \cos x - 1$ shifts the cosine graph 1 unit downwards.

Vertical shifts alter the midline of the trigonometric functions but do not affect their amplitude or period.

Amplitude, Period, Phase, and Vertical Shifts in Tangent

Unlike sine and cosine, the tangent function inherently has a different set of transformations due to its undefined points:

• Amplitude: Undefined.
• Period: Governed by $B$, with period $\frac{\pi}{B}$.
• Phase Shift: Shifted horizontally by $\frac{C}{B}$ units.
• Vertical Shift: Shifted vertically by $D$ units.

For example, $y = \tan(2x - \frac{\pi}{2}) + 1$ has:

• Period: $\frac{\pi}{2}$
• Phase Shift: $\frac{\pi}{4}$ units to the right
• Vertical Shift: 1 unit upwards

These transformations affect the location of the asymptotes and the general shape of the tangent graph.

Identifying Key Points

Key points are crucial for accurately sketching trigonometric graphs. These include:

• Intercepts: Points where the graph crosses the x-axis.
• Maximum and Minimum Points: Highest and lowest points on the graph.
• Asymptotes (for Tangent): Lines the graph approaches but never touches.

By calculating and plotting these points, students can create precise and detailed graphs.

Steps to Sketch Trigonometric Graphs

Following a systematic approach ensures accuracy:

1. Identify the Function: Determine whether it's sine, cosine, or tangent.
2. Determine Transformations: Note any amplitude, period, phase shift, or vertical shift.
3. Find Key Points: Calculate intercepts, maxima, minima, and asymptotes.
4. Plot Points: Graph the key points on the coordinate plane.
5. Draw the Curve: Connect the points smoothly, respecting the transformations.
6. Label Asymptotes (for Tangent): Draw dashed lines where the function approaches but never reaches.

Practicing these steps leads to proficiency in graphing trigonometric functions.

Applications of Trigonometric Graphs

Understanding these graphs is essential for modeling periodic phenomena such as sound waves, light waves, and oscillatory motion. In mathematics, they aid in solving equations and understanding function behavior. In physics, they are used to describe oscillations, waves, and circular motions.

Common Challenges and Solutions

Students often struggle with identifying the correct transformations and accurately plotting key points. To overcome these challenges:

• Practice: Regularly sketch various transformed functions to build confidence.
• Understand Transformations: Grasp how each parameter affects the graph.
• Use Technology: Graphing calculators and software can provide visual feedback.
• Check Work: Verify key points and transformation effects for accuracy.

Comparison Table

Summary and Key Takeaways