4.7 inverse trigonometric functions. for an inverse to exist the function must be one- to - one a...
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4.7 INVERSE TRIGONOMETRIC FUNCTIONS
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For an inverse to exist the function MUST be one- to - one
• A function is one-to-one if for every x there is exactly one y and for every y there is exactly one x.
• So• If x and/or y is raised
to an even power then the inverse does not exist unless the domain is restricted.
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• The equation y = x2
• does not have an inverse because two different x values will produce the same y-value.
• i.e. x = 2 and x = -2 will produce y = 4.
• The horizontal line test fails.
• In order to restrict the domain, a basic knowledge of the shape of the graph is crucial. This is a parabola with (0,0) as the vertex. Restrict the domain to the interval [0,infinity) to make it one-to-one.
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Now let’s look at the trig functions
x
y
x
y
x
y
y = sin x y = cos x
y = tan x
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x
y
For the graph of y = sin x, the Domain is (-∞, ∞) the Range is [-1, 1]
Not a 1-1 functionSo it currently does not have an inverse
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x
y
However we can restrict the domain to [- Note the range will remain [-1, 1]
Now it’s 1-1!
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x
y y = sinx
The inverse of sinx
or
Is denoted as arcsinx
x1sin
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On the unit circle:
x
y
For the inverse sine function with angles only from -toour answers will only be in either quadrant 1 for positive values and quadrant 4 for negative values.
Find the exact value, if possible,
1 -11 3arcsin sin sin 2
2 2
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x
y
y = cos x is not one to one, so its domain will also need to be restricted.
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y = cos x is not one to one, so its domain will also need to be restricted.
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x
y
On this interval, [0, ] the cosine function is one-to-one and we can now define the inverse cosine function.y = arccos x or y = cos-1 x
y = arccos x
y = cos x
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On the unit circle ,inverse cosine will only exist in quadrant 1 if the value is positive and quadrant 2 if the value is negative.
x
y
Find the exact value for:
-12 3arccos arccos( 1) cos
2 2
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y = tan x
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x
y
Remember that tangent is undefined at -and
y = tanx
y = arctanx
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x
y
Remember that tangent is undefined at -and
Find the exact value
1 3arctan 1 tan 0 arctan
3
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Using the calculator.
• Be in radian mode• Arctan(-15.7896)• Arcsin(.3456)• Arccos(-.6897)• Arcsin(1.4535)• Arccos(-2.4534)
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H Dub
• 4-7 Page 349 #1-16all, 49-67odd