Sec. 5.8 Inverse Trig Functions and Differentiation

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Sec. 5.8 Inverse Trig Functions and
Differentiation

• By
• Dr. Julia Arnold

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Since a function must pass the horizontal line
test to have an inverse function, the trig
functions, being periodic, have to have their
domains restricted in order to pass the
horizontal line test. For example Lets look at
the graph of sin x on -2?, 2 ?.
Flunks the horizontal line test.
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By restricting the domain from - ?/2, ?/2 we
produce a portion of the sine function which will
pass the horizontal line test and go from -1,1.
The inverse sine function is written as y
arcsin(x) which means that sin(y)x. Thus y is an
angle and x is a number.
Y sinx
Y arcsin x
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We must now do this for each of the other trig
functions
Y cos x on 0, ? Range -1,1
Y arccos (x) on -1,1 range 0, ?
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arctan(x) has a range of (- ?/2, ?/2) arccot(x)
has a range of (0, ?) arcsec(x) has a range of
0, ? y ?/2 arccsc(x) has a range of -
?/2, ?/2 y 0
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Evaluate without a calculator
Step 1 Set equal to x
Step 2 Rewrite as
Step 3 Since the inverse is only defined in
quadrants 1 4 for sin we are looking for an
angle in the 4th quadrant whose value is -1/2.
The value must be - ?/6.
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Evaluate without a calculator
Step 1 Set equal to x
Step 2 Rewrite as
Step 3 The inverse is only defined in quadrants
1 2 for cos so we are looking for the angle
whose value is 0.
The value must be ?/2.
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Evaluate without a calculator
Step 3 The inverse is only defined in quadrants
1 4 for tan so we are looking for the angle
whose tan value is ?3.
The tan 60 ?3 or x ?/3
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Evaluate with a calculator
Check the mode setting on your calculator.
Radian should be highlighted. Press 2nd function
sin .3 ) Enter. The answer is .3046
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Inverse Properties
If -1 lt x lt 1 and - ?/2 lt y lt ?/2 then
sin(arcsin x)x and arcsin(siny)y
If - ?/2 lt y lt ?/2 then tan(arctan x)x and
arctan(tany)y
If -1 lt x lt 1 and 0 lt y lt ?/2 or ?/2 lt y lt ?
then sec(arcsec x)x and arcsec(secy)y
On the next slide we will see how these
properties are applied
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Inverse Properties Examples
Solve for x
? This is y
If - ?/2 lt y lt ?/2 then tan(arctan x)x and
arctan(tany)y
Thus
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Inverse Properties Examples
If
where 0 lt y lt ?/2 find cos y
Solution For this problem we use the right
triangle
Sin(y) x, thus the opp side must be x and the
hyp must be 1, so sin y x
1
x
By the pythagorean theorem, this makes the bottom
side
Cos(y)
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Inverse Properties Examples
find tan y
If
Solution For this problem we use the right
triangle
By the pythagorean theorem, this makes the opp
side
1
2
tan(y)
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Derivatives of the Inverse Trig Functions
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Examples Using the Derivatives of the Inverse
Trig Functions
Note u du/dx
Find the derivative of arcsin 2x
Let u 2x du/dx 2
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Examples Using the Derivatives of the Inverse
Trig Functions
Note u du/dx
Find the derivative of arctan 3x
Let u 3x du/dx 3
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Examples Using the Derivatives of the Inverse
Trig Functions
Find the derivative of
Let
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For your next test make a 4x6 notecard and
copy all of the derivatives summarized on page
382.
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Some homework examples
22. Page 383
Let
Solution Use the right triangle on the
coordinate graph
Now using the triangle we can find sec x after
we find the hyp.
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Some homework examples
24. Page 383 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
48. Page 384 Find the derivative of
Let u
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On Wednesday there will be Quiz 3B on the
homework for 5.1, 5.2, 5.3, 5.4, 5.5, 5.8
While you are in BB, please go to
Communication then to discussion board and answer
the question I have posed. Thanks.
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This is where Ive been in Orlando, Fl at the
Swan Hotel on the Disney property.