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Lecture 6

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University of Miami

Mathematics

MTH 162

Pachero

Fall

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MTH 162
Calculus II
5.6 Inverse Trigonometric Functions Notes
L. Sterling
Abstract
Provide a generalization to each of the key terms listed in this section.
Functions and Their Inverses
What are the two main functions?
f ▯1(f (x)) = x
f f ▯1 (x) = x
What are the properties of those two functions?
▯ ▯
▯ f▯1 (f (x)) = x and f f1(x) = x
▯1
– f ▯ (f (x▯) = x would occur every x that’s in f’s domain.
– f f ▯1 (x) = x for every x that’s in f’s domain
▯ Domain and Range
▯1
– f’s domain = f ’s range
– f’s range = f▯1’s domain
▯1
▯ Both f’s and f ’s graph are symmetric, but with respect to the line y = x.
▯ If and Then
– If you have a function has its own inverse function, then the implicit equation of the
inverse function would be x = f(y).
– If you are solving the given equation for y, then you would have to obtain the explicit
equation that would look like y =(x).
Inverse Sine Function
▯1
y = sin (x)
x = sin(y)
▯1 ▯ x ▯ 1
▯ ▯
▯ ▯ y ▯
2 2
1 Properties
f ▯1(f (x)) = sin (sin(x)) = x
▯ ▯ ▯ x ▯ ▯
2 2
▯ ▯1 ▯ ▯ ▯1 ▯
f f (x) = sin sin (x) = x
▯1 ▯ x ▯ 1
Inverse Cosine Function
y = cos▯1(x)
x = cos(y)
▯1 ▯ x ▯ 1
0 ▯ y ▯ ▯
Properties
▯1 ▯1
f (f (x)) = cos (sin(x)) = x
0 ▯ x ▯ ▯
▯ ▯ ▯ ▯
f f ▯1(x) = sin sin ▯1(x) = x
▯1 ▯ x ▯ 1
Inverse Tangent Function
y = tan▯1(x)
x = tan(y)
▯1 ▯ x ▯ 1
▯ ▯ ▯ y ▯ ▯

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