Embed Size (px)
Citation preview
CH 10.1.notebook
1
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Later in Chapter 10, we will use sine and cosine graphs to create mathematical equations the can be used to learn about cell phones, tides, satellites, Ferris wheels and car engines.
First we need to learn about the characteristics of sine, cosine and tangent graphs.
CH 10.1.notebook
2
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The graph of the sine function is described as a wave.
CH 10.1.notebook
3
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The domain of the sine function is all real numbers. Which means sin(x) can have any input value of x.
CH 10.1.notebook
4
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The range of sin(x) is 1≤y≤1, which means that sin(x) has a minimum value of 1 and a maximum value of 1, just like it does on the unit circle.
CH 10.1.notebook
5
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
1
1
CH 10.1.notebook
6
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The cosine function has the same domain and range characteristics as the sine function.
1
1
CH 10.1.notebook
7
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
We are going to learn about THREE basic characteristics of the sine and cosine functions:
I Amplitude
II PeriodIII Frequency
CH 10.1.notebook
8
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The amplitude is the vertical distance from the top of the wave to the bottom of the wave divided by two.
Maximum Minimum
2
CH 10.1.notebook
9
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
1
1
CH 10.1.notebook
10
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
11
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
3
3
CH 10.1.notebook
12
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
13
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
14
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The amplitude for cosine is calculated in exactly the same way as it is for sine.
IF:
THEN:
CH 10.1.notebook
15
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The sine or cosine wave is not always centered on the xaxis. However, we can still calculate the amplitude. Later in this chapter we will learn how to translate periodic functions to create the following graph.
CH 10.1.notebook
16
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Max = 5
Min = 1
CH 10.1.notebook
17
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Maximum Minimum
2
CH 10.1.notebook
18
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The period of a function is the distance on the xaxis a function travels until it starts repeating itself.
A function is periodic if its graph posses a repeating pattern.
CH 10.1.notebook
19
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Period = 2π
CH 10.1.notebook
20
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
When:
CH 10.1.notebook
21
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
What is the period for the following function:
Pull
Pull
CH 10.1.notebook
22
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Period = π
CH 10.1.notebook
23
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
What is the period of this function?
CH 10.1.notebook
24
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
25
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
8π=25.13
CH 10.1.notebook
26
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
What is the period of this function?
CH 10.1.notebook
27
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
28
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
29
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
As the value of b increases, the graph of the function looks more compressed.
As the value of b decreases, the graph of the function looks more stretched out.
CH 10.1.notebook
30
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Find the b value of the function.
4π2π
π
3π
CH 10.1.notebook
31
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The period of the wave is 4π, so to find b, use the period formula.
CH 10.1.notebook
32
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
If b=½, now find the value of a and use that value to find the equation of the sine wave.
CH 10.1.notebook
33
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
34
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Find the b value of the function.
CH 10.1.notebook
35
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The period of the wave is , so to find b, use the period formula.
CH 10.1.notebook
36
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
If b=4 now find the value of a and use that value to find the equation of the cosine wave.
CH 10.1.notebook
37
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
38
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Frequency is the number of completed cycles per unit of time. Often referred to as cycles per second.
CH 10.1.notebook
39
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Finding xintercepts of sin(x)
0π
2π3π
4π5π-π-2π
-3π
CH 10.1.notebook
40
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The xintercepts of sin(x) are: 0, ±π, ±2π, ±3π, ...
Also written as kπ, where k is an integer.
CH 10.1.notebook
41
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Finding xintercepts of cos(x)
π 5π3π 7π
-π-5π-3π
2 22
22
2
2
CH 10.1.notebook
42
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
The xintercepts of cos(x) are:
Also written as , where k is an odd integer.
CH 10.1.notebook
43
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Finding the intercepts of:
where n is an integer.
CH 10.1.notebook
44
April 03, 2012
Graph Sine, Cosine, and Tangent FunctionsFind the intercepts of:
CH 10.1.notebook
45
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
46
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Now start substituting in values for n.
CH 10.1.notebook
47
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
You can also substitute in negative values for n.
CH 10.1.notebook
48
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Finding the intercepts of:
where n is an integer.
CH 10.1.notebook
49
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Find the intercepts of:
CH 10.1.notebook
50
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Now start substituting in values for n.
CH 10.1.notebook
51
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
52
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
53
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
54
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Asymptotes
CH 10.1.notebook
55
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
Characteristics of y=tan(x) graph• The domain is all real numbers except odd multiples of π/2. At these values, the graph has a vertical asymptote.• The range is all real numbers, therefore there is no minimum, maximum or amplitude.• The graph has a period of π.• The xintercepts occur when:
x = 0, ±π, ±2π, ±3π...
CH 10.1.notebook
56
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
57
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
58
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
As b becomes bigger, the graph becomes more densely packed together.
CH 10.1.notebook
59
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
60
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
61
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
As the value of a increases, the graph becomes more linear.
CH 10.1.notebook
62
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
What are the period and the asymptotes of this function?
CH 10.1.notebook
63
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
CH 10.1.notebook
64
April 03, 2012
Graph Sine, Cosine, and Tangent Functions
