Download However, if you restrict the domain toанπ/2 < x <аπ/2... y

4.7 Inverse Trigonometric Functions
Properties of Inverse Functions:
In order for a function to have an inverse function, it must be one ­ to ­ one, that is, it must pass the Horizontal Line Test.
Does the sine graph pass the Horizontal Line Test?
-π
π
2π
3π
4π
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However, if you restrict the domain to ­π/2 < x < π/2...
y = sin x has a unique inverse function:
y = arcsin x or y = sin­1x
inverse finds the angle (or arc) whose sine is x
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Find the exact value.
c.
d.
e.
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The Arccos Function
Does cosine have an inverse?
-π
π
2π
3π
4π
What should we restrict the domain to?
What is the range?
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2
The Arctan Function
tangent?
-π
π
2π
3π
4π
What is the domain?
What is the range?
Therefore, for arctan...
Domain:
Range:
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Inverse Trigonometric Functions
The inverse sine function is defined by...
y = arcsin x Domain: ­1< x < 1
Range: ­π/2 < y < π/2
The inverse cosine function is defined by...
y = arccos x Domain: ­1 < x < 1
Range: 0 < y < π
The inverse tangent function is defined by...
y = arctan x Domain: ­∞< x < ∞
Range: ­π/2 < y < π/2
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Examples: Find the exact value of:
arcsin (­1)
Reminders:
1. arc finds the angle
arccos ­1/2
2. what is the range?
arctan √3
arccos √2/2
cos ­1 ­1
arctan 0
tan ­1 ­1
arcsin ­√3/2
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Find the exact value of:
arcsin(sin 3π/4)
cos(arccos 0)
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Find the exact value of:
Draw triangles in the correct quadrant!
tan(arccos(2/3))
cos(arcsin(­3/5))
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Write as an algebraic expression in x:
sin(arccos(3x))
cot(arcsin(x))
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5
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