BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter C, Problem 27E
To determine

To calculate:

The value of following expression:

  sin(x+y)

Expert Solution

Answer to Problem 27E

The given expression sin(x+y) is evaluated as 4+6215 .

Explanation of Solution

Given information:

  sinx=13secy=540x,yπ2

Formula Used:

Addition formula:

  sin(x+y)=sinxcosy+cosxsiny

Calculation:

Know that:

  sinx=oppositehypotenussinx=13

First need to sketch the triangle for sinx :

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter C, Problem 27E , additional homework tip  1

Fig. 1 Triangle for sinx

So, hypotenuse is equal to 3 and the length of the adjacent side is equal to 1 . So, use the Pythagoras theorem:

  r2=opposite2+adjacent232=12+adjacent2adjacent2=8adjacent=22

Know that:

  secx=hypotenusadjecentsecx=54

Need to sketch the triangle for secx :

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter C, Problem 27E , additional homework tip  2

Fig. 2 Triangle for secx

So, hypotenuse is equal to 5 and the length of the adjacent side is equal to 4 . So, use the Pythagoras theorem:

  r2=opposite2+a252=opposite2+42opposite2=9opposite=3

And the remaining side opposite to y is 3 .

From first triangle:

  sinx=13cosx=223

From second triangle:

  siny=35cosy=45

Now, evaluate the given expression as:

  sin(x+y)=sinxcosy+cosxsinysin(x+y)=(13)(45)+(223)(35)sin(x+y)=415+6215sin(x+y)=4+6215

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