Explanation Given: Identity: sin x + sin y = 2 sin ( x + y 2 ) cos ( x − y 2 ) Inequality: | sin x x | ≤ 1 for x ≠ 0 To verify the relation sin ( a + h ) − sin a = h sin ( h 2 ) ( h 2 ) cos ( a + h 2 ) using the given identity sin x + sin y = 2 sin ( x + y 2 ) cos ( x − y 2 ) _ We have verify the relation sin ( a + h ) − sin a = h sin ( h 2 ) ( h 2 ) = cos ( a + h 2 ) by using the given identity sin x + sin y = 2 sin ( x + y 2 ) cos ( x − y 2 ) .For that first we have to derive the equation for sin x − sin y . Then we have to replace x by ( a + h ) and y by a . And to verify the relation we have to simplify the result which we obtain from the above step. Formula Used: The trigonometric identity: sin x + sin y = 2 sin ( x + y 2 ) cos ( x − y 2 ) . Calculation: Description sin ( a + h ) − sin a = h sin ( h 2 ) ( h 2 ) cos ( a + h 2 ) Step 1 : Write the given identity as itself. sin x + sin y = 2 sin ( x + y 2 ) cos ( x − y 2 ) Step 2 : We know that sin ( − x ) = − sin x . Thereby we substitute sin ( − y ) in the above identity to find sin x − sin y . sin x + sin ( − y ) = 2 sin ( x + ( − y ) 2 ) cos ( x − ( − y ) 2 ) A s sin ( − y ) = − sin y , sin x + sin ( − y ) = sin x − sin y sin x − sin y = 2 sin ( x − y ) 2 ) cos ( x + y ) 2 ) Step 3 : Now we have to find sin ( a + h ) − sin a by using the above identity. For that we substitute x = a + h a n d y = a in the above identity to find sin ( a + h ) − sin a . sin x − sin y = 2 sin ( x − y ) 2 ) cos ( x + y ) 2 ) sin ( a + h ) − sin a = 2 sin ( a + h − a 2 ) cos ( a + h + a 2 ) ⇒ 2 sin ( h 2 ) cos ( 2 a + h 2 ) = 2 sin ( h 2 ) cos ( 2 a 2 + h 2 ) ⇒ 2 sin ( h 2 ) cos ( a + h 2 ) Step 4 : Multiply and divide the right side of the equation by h 2 . sin ( a + h ) − sinh = 2 sin ( h 2 ) cos ( a + h 2 ) ⇒ ( h 2 ) 2 sin ( h 2 ) cos ( a + h 2 ) ( h 2 ) ⇒ h sin ( h 2 ) cos ( a + h 2 ) ( h 2 ) ⇒ h sin ( h 2 ) ( h 2 ) cos ( a + h 2 ) Step 5 : Verify the result obtained in the above equation with the give result. On verification we confirm that both are same. Thereby the given relation is verified. sin ( a + h ) − sin a = h sin ( h 2 ) ( h 2 ) cos ( a + h 2 ) To show that | sin ( a + h ) − sin a | < | h | _ for all a _ by using the given inequality | sin x x | ≤ 1 _ for x ≠ 0 _ Already we verified the relation sin ( a + h ) − sin a = h sin ( h 2 ) ( h 2 ) = cos ( a + h 2 ) . Thereby to show | sin ( a + h ) − sin a | < | h | , we use h sin ( h 2 ) ( h 2 ) + cos ( a + h 2 ) instead of sin ( a + h ) − sin a . Here they have given the inequality | sin x x | ≤ 1 . We have to substitute the obtained verified result to show that | sin ( a + h ) − sin a | < | h | . Formula Used: The trigonometric inequality: | sin x x | ≤ 1 . Calculation: Description | sin ( a + h ) − sin a | < | h | _ Step 1 : Write the given inequality as itself

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 2.9, Problem 29E

To determine

Verification of the relation sin(a+h)−sina=hsin(h2)(h2)=cos(a+h2) by using the given identity

To show that |sin(a+h)−sina|<|h| for all a by using the given inequality

Finally to prove rigorously that limx→asinx=sina

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See solution

Chapter 2.9, Problem 28E

Chapter 2.9, Problem 30E

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Chapter 2 Solutions

Calculus - Standalone book

Ch. 2.1 - Prob. 1PQ Ch. 2.1 - Prob. 2PQ Ch. 2.1 - Prob. 3PQ Ch. 2.1 - Prob. 4PQ Ch. 2.1 - Prob. 5PQ Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E

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Verification of the relation sin ( a + h ) − sin a = h sin ( h 2 ) ( h 2 ) = cos ( a + h 2 ) by using the given identity To show that | sin ( a + h ) − sin a | < | h | for all a by using the given inequality Finally to prove rigorously that lim x → a sin x = sin a

Solution Summary: The author explains how to determine the relation mathrmsin(a+h)-macros (raisebox1exh!left/

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Verification of the relation sin(a+h)−sina=hsin(h2)(h2)=cos(a+h2) by using the given identity

To show that |sin(a+h)−sina|<|h| for all a by using the given inequality

Finally to prove rigorously that limx→asinx=sina

Want to see the full answer?

Chapter 2 Solutions

Verification of the relation sin ( a + h ) − sin a = h sin ( h 2 ) ( h 2 ) = cos ( a + h 2 ) by using the given identity To show that | sin ( a + h ) − sin a | &lt; | h | for all a by using the given inequality Finally to prove rigorously that lim x → a sin x = sin a

Solution Summary: The author explains how to determine the relation mathrmsin(a+h)-macros (raisebox1exh!left/

Concept explainers

Videos

Verification of the relation sin(a+h)−sina=hsin(h2)(h2)=cos(a+h2) by using the given identity To show that |sin(a+h)−sina|<|h| for all a by using the given inequality Finally to prove rigorously that limx→asinx=sina

Want to see the full answer?

Chapter 2 Solutions

Verification of the relation sin ( a + h ) − sin a = h sin ( h 2 ) ( h 2 ) = cos ( a + h 2 ) by using the given identity To show that | sin ( a + h ) − sin a | < | h | for all a by using the given inequality Finally to prove rigorously that lim x → a sin x = sin a

Verification of the relation sin(a+h)−sina=hsin(h2)(h2)=cos(a+h2) by using the given identity

To show that |sin(a+h)−sina|<|h| for all a by using the given inequality

Finally to prove rigorously that limx→asinx=sina