Inverse Trig. Functions

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Inverse Trig. Functions Differentiation

• Section 5.8

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INVERSE TRIGONOMETRIC FUNCTIONS

• Here, you can see that the sine function y sin
  x is not one-to-one.
• Use the Horizontal Line Test.

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INVERSE TRIGONOMETRIC FUNCTIONS

• However, here, you can see that the function
  f(x) sin x, , is
  one-to-one.

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INVERSE SINE FUNCTIONS
Equation 1

• As the definition of an inverse function states
• that
• we have
• Thus, if -1 x 1, sin-1x is the number between
  and whose sine is x.

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INVERSE SINE FUNCTIONS
Example 1

• Evaluate
• a.
• b.

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Solve.
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Solve.
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Example 1 a
INVERSE SINE FUNCTIONS

• We have
• This is because , and
  lies between and .

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Example 1 b
INVERSE SINE FUNCTIONS

• Let , so .
• Then, we can draw a right triangle with angle ?.
• So, we deduce from the Pythagorean Theorem that
  the third side has length .

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INVERSE SINE FUNCTIONS
Example 1b

• This enables us to read from the triangle that

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INVERSE SINE FUNCTIONS
Equations 2

• In this case, the cancellation equations
• for inverse functions become

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INVERSE SINE FUNCTIONS

• The graph is obtained from that of
• the restricted sine function by reflection
• about the line y x.

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INVERSE SINE FUNCTIONS

• We know that
• The sine function f is continuous, so the
  inverse sine function is also continuous.
• The sine function is differentiable, so the
  inverse sine function is also differentiable
  (from Section 3.4).

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INVERSE SINE FUNCTIONS

• since we know that is sin-1 differentiable, we
  can just as easily calculate it by implicit
  differentiation as follows.

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INVERSE SINE FUNCTIONS

• Let y sin-1x.
• Then, sin y x and p/2 y p/2.
• Differentiating sin y x implicitly with respect
  to x,we obtain

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INVERSE SINE FUNCTIONS
Formula 3

• Now, cos y 0 since p/2 y p/2, so

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INVERSE SINE FUNCTIONS
Example 2

• If f(x) sin-1(x2 1), find
• the domain of f.
• f (x).

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INVERSE SINE FUNCTIONS
Example 2 a

• Since the domain of the inverse sine function is
  -1, 1, the domain of f is

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Example 2 b
INVERSE SINE FUNCTIONS

• Combining Formula 3 with the Chain Rule, we have

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INVERSE COSINE FUNCTIONS
Equation 4

• The inverse cosine function is handled similarly.
  
• The restricted cosine function f(x) cos x, 0
  x p, is one-to-one.
• So, it has an inverse function denoted by cos-1
  or arccos.

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INVERSE COSINE FUNCTIONS
Equation 5

• The cancellation equations are

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INVERSE COSINE FUNCTIONS

• The inverse cosine function,cos-1, has domain
  -1, 1 and range , and is a
  continuous function.

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INVERSE COSINE FUNCTIONS
Formula 6

• Its derivative is given by
• The formula can be proved by the same method as
  for Formula 3.

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INVERSE TANGENT FUNCTIONS

• The inverse tangent function, tan-1 arctan,
  has domain and range .

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INVERSE TANGENT FUNCTIONS

• We know that
• So, the lines are vertical asymptotes of the
  graph of tan.

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INVERSE TANGENT FUNCTIONS

• The graph of tan-1 is obtained by reflecting the
  graph of the restricted tangent function about
  the line y x.
• It follows that the lines y p/2 and y -p/2
  are horizontal asymptotes of the graph of
  tan-1.

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Inverse Trig. Functions

• None of the 6 basic trig. functions has an
  inverse unless you restrict their domains.

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• Function Domain Range
• y arcsin x -1
• y arccos x -1
• y arctan x
• y arccot x
• y arcsec x I II
• y arccsc I IV

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The Inverse Trigonometric Functions
Graphs of six inverse trigonometric functions
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The Inverse Trigonometric Functions
Graphs of six inverse trigonometric functions
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Inverse Properties

• f (f 1(x)) x and f 1(f (x)) x
• Remember that the trig. functions have inverses
  only in restricted domains.

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Table 11
DERIVATIVES

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Derivatives of Inverse Trig. Functions

• Let u be a differentiable function of x.

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DERIVATIVES

• Each of these formulas can be combined with the
  Chain Rule.
• For instance, if u is a differentiable function
  of x, then

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DERIVATIVES
Example 5

• Differentiate

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DERIVATIVES
Example 5 a

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DERIVATIVES
Example 5 b

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Find each derivative with respect to x.
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Find each derivative with respect to x.
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Find each derivative with respect to the given
variable.
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Find each derivative with respect to the given
variable.
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Example 1
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Example 2
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Example 2
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Example 2
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Example 2
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Example 2
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Example 2
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Example 2
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Some homework examples
Write the expression in algebraic form
Let then
Solution Use the right triangle
Now using the triangle we can find the hyp.
3x
y
1
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Some homework examples
Find the derivative of
Let u
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Example
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Example
55
Find an equation for the line tangent to the
graph of at x -1
Slope of tangent line
When x -1, y
At x -1