Precalculus: Mathematics for Calculus - 6th Edition
Precalculus: Mathematics for Calculus - 6th Edition
6th Edition
ISBN: 9780840068071
Author: Stewart, James, Redlin, Lothar, Watson, Saleem
Publisher: Cengage Learning
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Fundamentals of Trigonometric Identities
Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.
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Chapter 7.3, Problem 103E
To determine

To show: that sin2A+sin2B+sin2C=4sinAsinBsinC

Expert Solution & Answer
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Explanation of Solution

Given information:

  A,B and C are the angles in a triangle.

Calculation:

Start the left hand side of the given expression and transform it to the right hand side.

  L.H.S=sin2A+sin2B+sin2C .

Using the sum to product formula, sinx+siny=2sinx+y2⋅cosx−y2 , for the first two terms of the left hand side, get

  (sin2A+sin2B)+sin2C=2sin2A+2B2⋅cos2A−2B2+sin2C=2sin2(A+B)2⋅cos(A−B)+sin2C=2sin(A+B)⋅cos(A−B)+sin2C

The sum of three angles of the triangle is 180° .

Therefore,

  A+B+C=πA+B=π−C

Taking sine function on both sides, get

  sin(A+B)=sin(π−C)=sinC

On substituting sin(A+B)=sinC in the first term and using the formula for sin2C ,

  sin2C=2sinC⋅cosC in the second term, get

  (sin2A+sin2B)+sin2C=2sinC⋅cos(A−B)+2sinC⋅cosC

Factoring 2sinC from the right hand side, get

  (sin2A+sin2B)+sin2C=2sinC(cos(A−B)+cosC)

The sum of three angles of the triangle is 180°

Therefore,

  A+B+C=πC=π−(A+B)

Taking sine function on both sides, get

  cosC=cos(π−(A+B))=−cos(A+B)

On substituting cosC=−cos(A+B) in the right hand side of the equation,

  (sin2A+sin2B)+sin2C=2sinC(cos(A−B)+cosC) , get

  (sin2A+sin2B)+sin2C=2sinC(cos(A−B)−cos(A+B))

Using sum to product formula cosx−cosy=−2sinx+y2⋅sinx−y2 , get

  (sin2A+sin2B)+sin2C=2sinC(−2sinA−B+A+B2−sinA−B−A−B2)=2sinC(−2sinA⋅sin(−B))

Using even odd identify, sin(−B)=−sinB , get

  (sin2A+sin2B)+sin2C=2sinC((−2sinA)(−sinB))=4sinA⋅sinB⋅sinC

Hence, shown that,

  (sin2A+sin2B)+sin2C=4sinA⋅sinB⋅sinC .

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Chapter 7.3, Problem 102E
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Chapter 7.3, Problem 104E

Chapter 7 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities; Author: Mathispower4u;https://www.youtube.com/watch?v=OmJ5fxyXrfg;License: Standard YouTube License, CC-BY