Inverse Trigonometric Functions

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Inverse Trigonometric Functions
Section 5.7

• Objectives
• Understand and use the inverse sine function.
• Understand and use the inverse cosine function.
• Understand and use the inverse tangent function.
• Use a calculator to evaluate inverse
  trigonometric functions.
• Find exact values of composite functions with
  inverse trig functions.

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The Inverse Sine Function
The inverse sine function, denoted by sin-1, is
the inverse of the restricted sine function y
sin x, Thus, y
sin-1 x means sin y x, where
and 1 lt x lt 1. We read y sin-1 x
as y equals the inverse sine at x.
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Finding Exact Values of sin-1x

• Let ? sin-1 x.
• Rewrite step 1 as sin ? x.
• Use the exact values in the table to find the
  value of ? in -?/2 , ?/2 that satisfies
• sin ? x.

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Exact Values for sin ?
? Sin ?









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Example

• Find the exact value of sin-1(1/2)

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Practice 1

• Find the exact value of

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Practice 2
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The Inverse Cosine Function

• The inverse cosine function, denoted by
• cos-1, is the inverse of the restricted cosine
  function
• y cos x, 0 lt x lt ?.
• Thus,
• y cos-1 x means cos y x,
• where 0 lt y lt ? and 1 lt x lt 1.

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Exact Values for cos ?
? Cos ?









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Text Example
Find the exact value of cos-1 Solution Step 1
Let ? cos-1 x. Thus ? cos-1 We must find the
angle ?, 0 lt ? lt ?, whose cosine equals Step
2 Rewrite ? cos-1 x as cos ? x. We obtain
cos ?
Step 3 Use the exact values in the table to
find the value of ? in 0, ? that satisfies cos
? x. The table on the previous slide shows that
the only angle in the interval 0, ? that
satisfies cos ? is 5?/6. Thus, ?
5?/6
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Practice 3
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The Inverse Tangent Function

• The inverse tangent function, denoted by tan-1,
  is the inverse of the restricted tangent function
  
• y tan x, . Thus,
• y tan-1 x means tan y x,
• where and ? lt x lt ?.

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Exact Values for tan ?
? Tan ?









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Example
Find the exact value of
Solution Step 1
thus, We must find the angle
whose tangent equals Step 2

We obtain Step 3 Use the exact values in
the table to find the value of ? in
that satisfies tan ? x. The
table shows that the only angle in the
interval that satisfies the equation is

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Practice 4
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Inverse Properties

• The Sine Function and Its Inverse
• sin (sin-1 x) x for every x in the interval
  -1, 1.
• sin-1(sin x) x for every x in the interval
  -?/2,?/2.
• The Cosine Function and Its Inverse
• cos (cos-1 x) x for every x in the interval
  -1, 1.
• cos-1(cos x) x for every x in the interval
  0, ?.
• 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 the interval
  (-?/2,?/2).