Chapter Contents

• Graphs of the Trigonometric Functions
• 1. Graphs of y = a sin x and y = a cos x
• 2. Graphs of y = a sin bx and y = a cos bx
• Frequency of Music Notes
• 3. Graphs of y = a sin(bx + c) and y = a cos(bx + c)
• 4. Graphs of tan, cot, sec and csc
• 5. Applications of Trigonometric Graphs
• 6. Composite Trigonometric Graphs
• Biorhythm Graphs
• 7. Lissajous Figures
• Graphs of the Trigonometric Functions Problem Solver

• Comments, Questions?

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From the math blog...

3. Graphs of y = a sin(bx + c) and y = a cos(bx + c)

by M. Bourne

In this section, we meet the following 2 graph types:

• y = a sin(bx + c)
• y = a cos(bx + c)

Both b and c in these graphs affect the phase shift (or displacement), given by:

The phase shift is the amount that the curve is moved in a horizontal direction from its normal position. The displacement will be to the left if the phase shift is negative, and to the right if the phase shift is positive.

There is nothing magic about this formula. We are just solving the expression in brackets for zero; bx + c = 0.

Example 1

Sketch the curve

y = sin(2x + 1).

Answer

Example 2

Sketch

Answer

Flash Interactive

In this Flash interactive, drag the curve left or right using the arrows. Observe the "c" and "displacement" values and how they change when you move the curve.

The period here is 2π/π = 2.

Phase Angle and Phase Shift

Phase angle is not always defined the same as phase shift.

The phase angle for the sine curve y = a sin(bx + c) is usually taken to be the value of c and the phase shift is usually given by -c/b, as we saw above.

Reminder: In the last section, we saw how to express sine curves in terms of frequency.

Example: Electronics engineers separate the terms "phase angle" and "phase shift", and they use a mix of radians and degrees. We may have a current expressed as follows:

I = 50 sin (2π(100)t + 30°)

This means the amplitude is 50 A, the frequency is 100 Hz and the phase angle is 30°.

See an application of phase angle at An Application to AC Circuits in the complex numbers chapter.

Also, see a discussion on this issue at Phase shift or phase angle? in the math blog.

To keep things simple for now, we will mostly use the term phase shift in this chapter.

Sine Graph Java Applet - Phase shift

The following applet will help you to understand the following graph sketching exercise.

In this Java applet, you can vary each of the amplitude, period and phase shift by using the sliders at the bottom.

You can also change the function to whatever you like. Try changing it to a*cos(b*x+c) and then vary the values of a, b or c with the sliders.

Show java applet

Exercises

Sketch the graph of the following.

1. y = sin(2x + π/6)

Answer

2. y = 3 sin(x + π/4)

Answer

3. y = 2 cos(x - π/8)

Answer

4. y = -cos(2x - π)

Answer

LiveMath Challenge

So, you think you've got the idea?

Click here for a challenge using LiveMath:

LIVEMath