Sec. 6.6: Inverse Trigonometric...
Transcript of Sec. 6.6: Inverse Trigonometric...
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Sec. 6.6: Inverse Trigonometric Functions
In this section, we will
I revisit definitions of trig functions - both right triangle andunit circle definitions
I define inverse trig functions and their domains and ranges
I revisit the cancellation laws given a function and its inversefunction
I find the derivatives of the inverse trig functions
I look at integral formulas of certain functions whose integralsare inverse trig functions
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Definition of trig functions and the ratios of the sides
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Unit circle definition
x = cos θ, y = sin θ
Example 1) Identify the corresponding (x , y) on the U.C. if θ = π6
Example 2) Find (x , y) if θ = 11π6
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Graph of the Sine function
Here’s the graph of the sine function.
Since the function is not 1-1, we need to restrict its domain todefine its inverse function. One way it can be done is by restringthe domain to [−π/2, π/2]. That is, let
D(sin x) = [−π/2, π/2]
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The graph of the inverse sine function
Here’s the graph of the sine function whose domain restricted to[−π/2, π/2].
For the above function, state its domain and range.
Recall that for the inverse function, we flip everything about y = x .
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The graph of the inverse sine function (Continued)
What are the domain and the range of the inverse sine function?
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Graph of the Cosine function
Here’s the graph of the cosine function.
To define its inverse function, we will need to restrict its domain,so that the cosine function becomes 1-1. One way it can be doneis by restring the domain to [0, π]. So, let
D(cos x) = [0, π].
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The graph of the inverse cosine functionHere’s the graph of the cosine function with its domain restrictedto [0, π].
For the above function, what is its domain and range?
Recall that for the inverse function, we flip everything about y = x .
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The graph of the inverse cosine function (Continued)
What are the domain and the range of the inverse cosine function?
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Cancellation equationsGiven f (x) with its domain D(f (x)), if g(x) = f −1(x), then
I f −1(f (x)) = x for x ∈ D(f )I f (f −1(x)) = x for x ∈ D(f −1)
Examples) Recall that
1. e(ln x) = x for x ∈ D(ln x)
2. ln(ex) = x for x ∈ D(ex)
3. a( ) = x for x ∈
4. loga( ) = x for x ∈
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Cancellation equations for the inverse sine function
Theorem
1. sin−1(sin x) = x , −π2 ≤ x ≤ π
2
2. sin(sin−1 x) = x , −1 ≤ x ≤ 1
Example) Evaluate 1) sin−1(1/2), 2) tan(arcsin 13), 3) sin−1(2).
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Cancellation equations for the inverse cosine function
Theorem
1. cos−1(cos x) = x , 0 ≤ x ≤ π2. cos(cos−1 x) = x , −1 ≤ x ≤ 1
Example) Evaluate 1) cos−1(1/2), 2) tan(arccos 14), 3)
cos−1(−2).
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Graph of the tangent function
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Graph of the inverse tangent function
1. What are the domain and the range of the inverse tangentfunction?
2. limx→∞ tan−1(x) =3. limx→−∞ tan−1(x) =
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Derivatives of the inverse trig functions : arcsine
Theorem
d
dxsin−1(x) =
1√1− x2
, −1 < x < 1
Example) Given f (x) = sin−1(2x), find
1. Domain of f2. f
′(x)
3. Domain of f′
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Derivatives of the inverse trig functions: arccosine
Theorem
d
dxcos−1(x) = − 1√
1− x2, −1 < x < 1
Example) Given f (x) = cos−1(x2 − 1), find
1. Domain of f2. f
′(x)
3. Domain of f′
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Derivatives of the inverse trig functions: arctangent
Theorem
d
dxtan−1(x) =
1
1 + x2, −∞ < x <∞
Example) Given f (x) = tan−1(3x), find
1. Domain of f2. f
′(x)
3. Domain of f′
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Integral formulas
Theorem
1.∫
1√1−x2 dx = sin−1 x + C
2.∫− 1√
1−x2 dx = cos−1 x + C
3.∫
11+x2
dx = tan−1 x + C
Examples) Evaluate
1.∫ 1/20
1√1−x2 dx
2.∫ 1/40
1√1−4x2 dx
3.∫ 10
41+x2
dx
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Examples1. Verify that ∫
1
a2 + x2dx =
1
atan−1(
x
a) + C
2. Find∫
19+x2
dx
3. Find∫ 30
19+x2
dx
4. Find∫
x9+x4
dx
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