Precalculus with Limits: A Graphing Approach
Precalculus with Limits: A Graphing Approach
7th Edition
ISBN: 9781305071711
Author: Ron Larson
Publisher: Brooks/Cole
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Trigonometric Identities
Trigonometry in mathematics deals with the right-angled triangle’s angles and sides. By trigonometric identities, we mean the identities we use whenever we need to express the various trigonometric functions in terms of an equation.
Inverse Trigonometric Functions
Inverse trigonometric functions are the inverse of normal trigonometric functions. Alternatively denoted as cyclometric or arcus functions, these inverse trigonometric functions exist to counter the basic trigonometric functions, such as sine (sin), cosi…
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Chapter 5.5, Problem 99E
To determine

To find: The value of the given function in two ways, using the given figure and trigonometric identities.

Expert Solution & Answer
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Answer to Problem 99E

Value of the given function is 413

Explanation of Solution

Given information:

The following figure:

  Precalculus with Limits: A Graphing Approach, Chapter 5.5, Problem 99E

The following function:

  sinαcosβ

Formula used:

The following Pythagorean identity:

  sin2θ+cos2θ=1

The following Pythagorean relation:

  Hypotenuse2=Adjacent2+Opposite2cosθ=AdjacentHypotenusesinθ=OppositeHypotenuse

Calculation:

From the given figure,

  Adjacent=12                                             [For α]Opposite=5        [For α]Hypotenuse2=Adjacent2+Opposite2       [Pythagorean relation]Hypotenuse2=(12)2+(5)2        [Substitute values]Hypotenuse2=144+25        [Find square]Hypotenuse2=169        [Add]Hypotenuse=13        [Find square root of both sides]cosα=AdjacentHypotenuse=1213        [Pythagorean relation]sinα=OppositeHypotenuse=513        [Pythagorean relation]Adjacent=4                                               [For β]Opposite=3        [For β]Hypotenuse2=Adjacent2+Opposite2       [Pythagorean relation]Hypotenuse2=(4)2+(3)2        [Substitute values]Hypotenuse2=16+9                    [Find square]Hypotenuse2=25        [Add]Hypotenuse=5        [Find square root of both sides]cosβ=AdjacentHypotenuse=45        [Pythagorean relation]sinβ=OppositeHypotenuse=35        [Pythagorean relation]

Value of sinαcosβ from the figure is calculated as:

  sinαcosβ=513⋅45 [Substitute]    =413           [Simplify]

Value of sinαcosβ from trigonometric identities is calculated as:

  sinαcosβ=1−cos2α⋅1−sin2β                            [Pythagorean identity]       =1−(1213)2⋅1−(35)2    [Substitute]       =1−144169.1−925                 [Square]        =25169.1625    [Subtract]      =513⋅45    [Find square root]      =413    [Simplify]

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Chapter 5.5, Problem 98E
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Chapter 5.5, Problem 100E

Chapter 5 Solutions

Precalculus with Limits: A Graphing Approach

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Inverse Trigonometric Functions; Author: Professor Dave Explains;https://www.youtube.com/watch?v=YXWKpgmLgHk;License: Standard YouTube License, CC-BY