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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.
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The graph of the sine function is described as a wave.
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The domain of the sine function is all real numbers. Which means sin(x) can have any input value of x.
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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.
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1
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The cosine function has the same domain and range characteristics as the sine function.
1
1
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We are going to learn about THREE basic characteristics of the sine and cosine functions:
I Amplitude
II PeriodIII Frequency
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The amplitude is the vertical distance from the top of the wave to the bottom of the wave divided by two.
Maximum Minimum
2
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1
1
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The amplitude for cosine is calculated in exactly the same way as it is for sine.
IF:
THEN:
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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.
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Max = 5
Min = 1
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Maximum Minimum
2
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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.
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Period = 2π
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When:
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What is the period for the following function:
Pull
Pull
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Graph Sine, Cosine, and Tangent Functions
Period = π
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What is the period of this function?
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8π=25.13
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What is the period of this function?
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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.
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Graph Sine, Cosine, and Tangent Functions
Find the b value of the function.
4π2π
π
3π
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The period of the wave is 4π, so to find b, use the period formula.
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If b=½, now find the value of a and use that value to find the equation of the sine wave.
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Find the b value of the function.
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The period of the wave is , so to find b, use the period formula.
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If b=4 now find the value of a and use that value to find the equation of the cosine wave.
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Frequency is the number of completed cycles per unit of time. Often referred to as cycles per second.
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Finding xintercepts of sin(x)
0π
2π3π
4π5π-π-2π
-3π
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The xintercepts of sin(x) are: 0, ±π, ±2π, ±3π, ...
Also written as kπ, where k is an integer.
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Finding xintercepts of cos(x)
π 5π3π 7π
-π-5π-3π
2 22
22
2
2
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The xintercepts of cos(x) are:
Also written as , where k is an odd integer.
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Finding the intercepts of:
where n is an integer.
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Graph Sine, Cosine, and Tangent FunctionsFind the intercepts of:
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Now start substituting in values for n.
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You can also substitute in negative values for n.
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Finding the intercepts of:
where n is an integer.
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Find the intercepts of:
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Now start substituting in values for n.
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Asymptotes
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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...
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As b becomes bigger, the graph becomes more densely packed together.
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As the value of a increases, the graph becomes more linear.
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What are the period and the asymptotes of this function?
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