Mat 161 PreCalculus

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Mat 161 - PreCalculus

• The Inverse Trigonometric Functions
• Sections 5.7

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The Inverse Sine Function
Consider the function f(x) sin (x) on its
natural domain. We know that it fails the HLT, so
it is not 1-1, therefore it is not invertible.
However, if we restrict the domain we can define
the inverse of sine.
Def The inverse sine function, denoted by arcsin
or sin-1 , is the inverse of the restricted sine
function y sin x, with domain - ?/2, ?/2 and
range -1, 1. Thus, y sin-1 x
means sin y x where - ?/2 y ?/2
and -1 x 1.
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The Inverse Sine Function
Arcsin x is the unique angle ? in the interval -
?/2, ?/2 So that sin ? x. So, in order for
this to be defined we must make sure that -1 x
1.
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The Inverse Sine Function
?/2

• Example Find the exact value of the following
• sin-1 (0)
• sin-1 (1/2)
• sin-1 (-v2/2)
• sin-1 (-1)
• sin-1 (30)

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The Inverse Cosine Function
Consider the function f(x) cos (x) on its
natural domain. We know that it fails the HLT, so
it is not 1-1, therefore it is not invertible.
However, if we restrict the domain we can define
the inverse of cosine.
Def The inverse cosine function, denoted by
arccos or cos-1 , is the inverse of the
restricted cosine function y cos x, with
domain 0, ? and range -1, 1. Thus,
y cos-1 x means cos y x where 0
y ? and -1 x 1.
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The Inverse Cosine Function
Arccos x is the unique angle ? in the interval
0, ? So that cos ? x. So, in order for this
to be defined we must make sure that -1 x 1.
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The Inverse Cosine Function

• Example Find the exact value of the following
• cos-1 (1)
• cos-1 (-1/2)
• cos-1 (v2/2)
• cos-1 (0)
• cos-1 (-3)

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The Inverse Tangent Function
Consider the function f(x) tan (x) on its
natural domain. We know that it fails the HLT, so
it is not 1-1, therefore it is not invertible.
However, if we restrict the domain we can define
the inverse of tangent.
Def The inverse tangent function, denoted by
arctan or tan-1 , is the inverse of the
restricted tangent function y tan x, with
domain (- ?/2, ?/2) and range (-8, 8). Thus,
y tan-1 x means tan y
x where - ?/2 lt y lt ?/2 and x is
any real number.
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The Inverse Tangent Function
Arcsin x is the unique angle ? in the interval (-
?/2, ?/2) So that tan ? x, where x can be any
real number.
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The Inverse Tangent Function

• Example Find the exact value of the following
• tan-1 (0)
• tan-1 (1/v3)
• tan-1 (-1)
• tan-1 (-v3)
• tan-1 (100)

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Use Calculators to evaluateThe Inverse Functions

• Example Find the value of the following using a
  calculator
• in radians to 7 decimal places
• sin-1 (-1/4)
• sin-1 (2/9)
• cos-1 (-1/3)
• cos-1 (-v2)
• tan-1 (0.00007)

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Cancelation Laws for Trigonometric Functions

• Inverse Properties
• The sine function and its inverse
• sin(sin-1 x) x for every x in -1,1
• sin-1(sin x) x for every x in -?/2, ?/2
• 2) The cosine function and its inverse
• cos(cos-1 x) x for every x in -1,1
• cos-1(cos x) x for every x in 0, ?
• 3) The tangent function and its inverse
• tan(tan-1 x) x for every real number x
• tan-1(tan x) x for every x in (-?/2, ?/2)

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Trigonometric Functions

• Find the exact value of each expression
• cos(cos-1 0.57)
• sin(sin-1 (-1/4))
• tan(tan-1 25)
• cos-1(cos 2?/3)
• sin-1(sin 3?/4)
• tan-1(tan 2?/3)

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Trigonometric Functions

• Find the exact value of each of the
• following composite expressions.
• sin(tan-1 1/3)
• sec(sin-1 (-1/10)
• csc(cos-1 (-3/5))

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Trigonometric Functions

• Simplify the following composite
• expressions. Assume x is positive and that
• the given expression is defined for x.
• tan(sin-1 (1/x))
• cot(tan-1 (3x)
• sin(cos-1 vx/4)

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Trigonometric Functions

• Reference
• Algebra and Functions
• Blitzer
• 3rd Edition