in this section, we will introduce the inverse trigonometric functions and construct their...
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In this section, we will introduce the inverse trigonometric functions and construct their derivative formulas.
Section 3.4 Inverse Trigonometric Functions
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Recall
A function is called one-to-one if whenever
it must be true that .
That is, an output value cannot come from two different input values.
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Recall
A function is called one-to-one if whenever
it must be true that .
That is, an output value cannot come from two different input values.
This function is not one-to-one.
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Recall
A function is called one-to-one if whenever
it must be true that .
That is, an output value cannot come from two different input values.
Significance:
Only one-to-one functions have inverse functions.
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Inverse Sine Function
Conceptual Idea
Below is shown the graph of
This function is not one-to-one and so has no inverse function.
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Inverse Sine Function
Conceptual Idea
Below is shown the graph of
This function has an inverse.
Consider restricting the domain of the sine function to:
This is the function in blue shown to the left.
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Inverse Sine Function
Definition
The function is the inverse of the sine function with restricted domain .
That is, the arcsin(x) is the angle θ in the interval
with .
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Inverse Cosine Function
Conceptual Idea
Below is shown the graph of
This function is not one-to-one and so has no inverse function.
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Inverse Cosine Function
Conceptual Idea
Below is shown the graph of
This function has an inverse.
Consider restricting the domain of the cosine function to:
This is the function in blue shown to the left.
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Inverse Cosine Function
Definition
The function is the inverse of the cosine function with restricted domain .
That is, the arccos(x) is the angle θ in the interval
with .
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Inverse Tangent Function
Conceptual Idea
Below is shown the graph of
This function is not one-to-one and so has no inverse function.
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Inverse Tangent Function
Conceptual Idea
Below is shown the graph of
This function has an inverse function.
Consider restricting the domain of the tangent function to:
This is the function in blue shown to the left.
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Inverse Tangent Function
Definition
The function is the inverse of the tangent function with restricted domain .
That is, the arctan(x) is the angle θ in the interval
with .
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Example 1
Use the definitions of the section to find the exact
value of .
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Example 2
Use the definitions of the section to find the exact
value of .
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Example 3
Use the definitions of the section to find the exact
value of .
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Example 4
Use the definitions of the section to find the exact
value of .
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TheoremDerivatives of Inverse Trig.
Functions
The following are true:
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TheoremDerivatives of Inverse Trig.
Functions
The following are true:
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Example 5
Find the derivative of the function .
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Example 6
Find the derivative of the function .
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Example 7
Find the derivative of the function .