Lecture Notes

There are three lecture notes for this section. Please click to open the lecture notes.

1. Basic sine and cosine functions (video lecture)(CC)

2. Sine and cosine functions with vertical or horizontal stretches/compressions or reflections (Video lecture) (CC)

3. The general form of sine and cosine functions with phase shifts (Video lecture) (CC)
Learning Objectives:

After you complete this section, you will be able to

1. describe the domain, range, amplitude, period and phase shifts of sine and cosine functions

2. draw the graphs of sine and cosine functions.

Study tips:

1. You will be able to understand the lecture notes easier after you watch the video clips and try to draw the graphs of functions by yourself. Watch the video clips first!

2. In the lecture note 'Basic sine and cosine functions', the first and the second page explains how to understand sine and cosine as 'functions'. If you have difficulties in understanding, then all you need to remember is 'sine and cosine functions are functions'. You can safely start the section from the useful facts on the third page.

3. Problems about the graphs on the WebAssign are multiple-choice problems. Here is the steps to take to choose the correct graph. (Return to this Study Tips when you solve the problems in WebAssign.)

a. Identify the graphs by looking at the pattern.

sine and cosine graphs.JPG

b. Identify the graph which has the correct amplitude.

c. Identify the graph which has the correct period.

d. Identify the graph which has the correct maximum and minimum points.

e. Identify the graph which has the correct horizontal shift.

Key Facts to remember:

In LaTeX: y=a sin [k(x-b)]+c and LaTeX: y= a cos[k(x-b)]+c , where LaTeX: b>0 , amplitude is $LaTeX: \vert a\vert$ , period is $LaTeX: \frac{2\pi}{k}\:$ , and the horizontal shift is .

Video Resources and Preliminary Exercise Problems:

Graphing Sine and Cosine With Different Coefficients, (Amplitude and Period), Ex 1 (CC)

Exercise:

Find the amplitudes and periods of the functions $LaTeX: y=-\cos\left(3x\right),\:y=\frac{1}{2}\sin x$ , and $LaTeX: y=\sqrt{2}\cos\left(\frac{1}{2}x\right)$ .

Graphing y = -2 cos(2x) (CC)Exercise:

Find the amplitude and period of the function $LaTeX: y=-2\cos\left(2x\right)$ . And then, graph the function of $LaTeX: y=-2\cos\left(2x\right)$ .

Graphing Sine and Cosine with Phase (Horizontal) Shifts, Example 1 (CC)

Exercise:

What are the phase shifts (horizontal shifts) of the functions $LaTeX: y=\cos\left(x-4\right),\:y=\sin\left[2\left(x+1\right)\right]$ , and $LaTeX: y=\cos\left(10x+30\right)$ ?

Graphing Sine and Cosine with Phase (Horizontal) Shifts, Example 2 (CC)

Exercise:

Find the amplitude, period, and phase shift (horizontal shift) of $LaTeX: y=-2\sin\left(x-\pi\right)$ . And then graph $LaTeX: y=-2\sin\left(x-\pi\right)$ .

Trigonometric Functions and Graphing: Amplitude, Period, Vertical and Horizontal Shifts, Ex 2 (CC)

Exercise:

Find the amplitude, period, and phase shift (horizontal shift) of $LaTeX: y=2\cos\left(x-\frac{\pi}{2}\right)+1$ . And then, graph $LaTeX: y=2\cos\left(x-\frac{\pi}{2}\right)+1$ .