Lesson Addition and subtraction of trigonometric functions

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<H2>Addition and subtraction of trigonometric functions - Examples</H2>

The addition and subtraction trigonometric functions formulas are:

{{{sin(alpha) + sin(beta) = 2*sin((alpha+beta)/2)*cos((alpha-beta)/2)}}}

{{{sin(alpha) - sin(beta) = 2*sin((alpha-beta)/2)*cos((alpha+beta)/2)}}}

{{{cos(alpha) + cos(beta) = 2*cos((alpha+beta)/2)*cos((alpha-beta)/2)}}}

{{{cos(alpha) - cos(beta) = -2*sin((alpha+beta)/2)*sin((alpha-beta)/2)}}}

{{{tan(alpha) +- tan(beta) = sin(alpha +- beta)/(cos(alpha)*cos(beta))}}}

{{{cot(alpha) +- cot(beta) = sin(alpha +- beta)/(sin(alpha)*sin(beta))}}}

The proofs of these formulas are presented in the lesson <A HREF= http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-of-trigonometric-functions.lesson> Addition and subtraction of trigonometric functions</A> in this module.
Below are examples of application of these formulas.

<B>Example 1</B>

Find sin(75°) + sin(15°).

<B>Solution</B>

Use the addition formula for sines:
{{{sin(alpha) + sin(beta) = 2*sin((alpha+beta)/2)*cos((alpha-beta)/2)}}}.

You have

sin(75°) + sin(15°) = 2*sin((75°+15°)/2)*cos((75°-15°)/2) = 2*sin(45°)*cos(30°) = {{{2*sqrt(2)/2*sqrt(3)/2 = sqrt(6)/2}}}.

<B>Example 2</B>

Prove that {{{(sin(alpha)+sin(3alpha))/(cos(alpha)+cos(3alpha)) = tan(2alpha)}}}.

<B>Solution</B>

Use the addition formula for sines and cosines:
{{{sin(alpha) + sin(beta) = 2*sin((alpha+beta)/2)*cos((alpha-beta)/2)}}},
{{{cos(alpha) + cos(beta) = 2*cos((alpha+beta)/2)*cos((alpha-beta)/2)}}}.

You have

{{{(sin(alpha)+sin(3alpha))/(cos(alpha)+cos(3alpha)) = 2*sin((alpha+3alpha)/2)*cos((alpha-3alpha)/2)/(2*cos((alpha+3alpha)/2)*cos((alpha-3alpha)/2)))}}} = {{{2*sin(2alpha)*cos(-2alpha)/(2*cos(2alpha)*cos(-2alpha)) = sin(2alpha)/cos(2alpha) = tan(2alpha)}}}.

The proof is completed.

<B>Example 3</B>

Prove that {{{1 + cos(2alpha) + cos(4alpha) + cos(6alpha) = 4*cos(alpha)*cos(2alpha)*cos(3alpha)}}}.

<B>Solution</B>

Using the addition formula for cosines you have
{{{1 + cos(2alpha) = cos(0) + cos(2alpha) = 2*cos((0+2alpha)/2)*cos((0-2alpha)/2) = 2*cos^2(alpha)}}},

{{{cos(4alpha) + cos(6alpha) = 2*cos((4alpha+6alpha)/2)*cos((4alpha-6alpha)/2) = 2*cos(5alpha)*cos(-alpha) = 2*cos(alpha)*cos(5alpha)}}}.

By summing the left and the right sides of these two equalities, you get
{{{1 + cos(2alpha) + cos(4alpha) + cos(6alpha) = 2*cos^2(alpha) + 2*cos(alpha)*cos(5alpha)}}}.

Furthermore, you can transform the right side as follows:
{{{2*cos^2(alpha) + 2*cos(alpha)*cos(5alpha) = 2*cos(alpha)*(cos(alpha) + cos(5alpha))}}}

= {{{2*cos(alpha)*(cos(alpha+5alpha)/2)*(cos(alpha-5alpha)/2)}}} = {{{2*cos(alpha)*cos(3alpha)*cos(-2alpha) = 2*cos(alpha)*cos(2alpha)*cos(3alpha)}}}.

The proof is completed.

<B>Example 4</B>

Prove yourself that {{{1 - cos(2alpha) + cos(4alpha) - cos(6alpha) = 4*sin(alpha)*cos(2alpha)*sin(3alpha)}}}.

<B>Solution</B>

The proof is similar to that of the <B>Example 3</B>.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For your convenience, below is the list of my lessons on Trigonometry in this site in the logical order. 
They all are under the current topic <B>Trigonometry</B> in the section <B>Algebra II</B>.

<B>Addition and subtraction formulas</B>
<TABLE>
  <TR>
  <TD>{{{cos(alpha + beta) = cos(alpha)*cos(beta) - sin(alpha)*sin(beta)}}},
{{{cos(alpha - beta) = cos(alpha)*cos(beta) + sin(alpha)*sin(beta)}}},

{{{sin(alpha + beta) = sin(alpha)*cos(beta) + cos(alpha)*sin(beta)}}},
{{{sin(alpha - beta) = sin(alpha)*cos(beta) - cos(alpha)*sin(beta)}}},

{{{tan(alpha + beta) = (tan(alpha) + tan(beta))/(1 - tan(alpha)*tan(beta))}}}, {{{tan(alpha - beta) = (tan(alpha) - tan(beta))/(1 + tan(alpha)*tan(beta))}}}.
 </TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;The lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson> Addition and subtraction formulas</A> and 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas-Examples.lesson>Addition and subtraction formulas - Examples</A>

</TD>
 </TR>
</TABLE>
<B>Addition and subtraction of trigonometric functions</B>
<TABLE>
  <TR>
  <TD>{{{sin(alpha) + sin(beta) = 2*sin((alpha+beta)/2)*cos((alpha-beta)/2)}}},

{{{sin(alpha) - sin(beta) = 2*sin((alpha-beta)/2)*cos((alpha+beta)/2)}}},

{{{cos(alpha) + cos(beta) = 2*cos((alpha+beta)/2)*cos((alpha-beta)/2)}}},

{{{cos(alpha) - cos(beta) = -2*sin((alpha+beta)/2)*sin((alpha-beta)/2)}}},

{{{tan(alpha) +- tan(beta) = sin(alpha +- beta)/(cos(alpha)*cos(beta))}}}, {{{cot(alpha) +- cot(beta) = sin(alpha +- beta)/(sin(alpha)*sin(beta))}}}.
 </TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;The lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-of-trigonometric-functions.lesson>Addition and subtraction of trigonometric functions</A> and 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Addition and subtraction of trigonometric functions - Examples</B> (this lesson)

</TD>
 </TR>
</TABLE>
<B>Product of trigonometric functions</B>
<TABLE>
  <TR>
  <TD>{{{sin(alpha)*sin(beta) = (1/2)*(cos(alpha-beta) - cos(alpha+beta))}}},

{{{cos(alpha)*cos(beta) = (1/2)*(cos(alpha-beta) + cos(alpha+beta))}}},

{{{sin(alpha)*cos(beta) = (1/2)*(sin(alpha-beta) + sin(alpha+beta))}}}.
 </TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Product-of-trigonometric-functions.lesson> Product of trigonometric functions</A> and 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Product-of-trigonometric-functions-Examples.lesson>Product of trigonometric functions - Examples</A>

</TD>
 </TR>
</TABLE>
<B>Powers of trigonometric functions</B>
<TABLE>
  <TR>
  <TD>{{{cos^2(alpha) = (1/2)*cos(2alpha) + 1/2}}},

{{{sin^2(alpha) = -(1/2)*cos(2alpha) + 1/2}}},

{{{cos^3(alpha) = (1/4)*cos(3alpha) + (3/4)*cos(alpha)}}},

{{{sin^3(alpha) = -(1/4)*sin(3alpha) + (3/4)*sin(alpha)}}}.
 </TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Powers-of-trigonometric-functions.lesson> Powers of Trigonometric functions</A> and 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Powers-of=trigonometric-functions-Examples.lesson>Powers of Trigonometric functions - Examples</A>

</TD>
 </TR>
</TABLE>
<B>Trigonometric functions of multiply argument</B>
<TABLE>
  <TR>
  <TD>{{{cos(2alpha) = 2*cos^2(alpha) - 1}}},

{{{sin(2alpha) = 2*sin(alpha)*cos(alpha)}}},

{{{cos(3alpha) = 4*cos^3(alpha) - 3*cos(alpha)}}},

{{{sin(3alpha) = -4*sin^3(alpha) + 3*sin(alpha)}}}.
 </TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-multiply-argument.lesson> Trigonometric functions of multiply argument</A> and 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-multiply-argument-Examples.lesson>Trigonometric functions of multiply argument - Examples</A>

</TD>
 </TR>
</TABLE>
<B>Trigonometric functions of half argument</B>
<TABLE>
  <TR>
  <TD>{{{sin^2(alpha/2) = (1-cos(alpha))/2}}}, {{{cos^2(alpha/2) = (1+cos(alpha))/2}}},

{{{tan(alpha/2) = sin(alpha)/(1+cos(alpha)) = (1-cos(alpha))/sin(alpha)}}},

{{{sin(alpha) = 2*tan(alpha/2)/(1+tan^2(alpha/2))}}}, {{{cos(alpha) = (1-tan^2(alpha/2))/(1+tan^2(alpha/2))}}}, {{{tan(alpha) = 2*tan(alpha/2)/(1-tan^2(alpha/2))}}}.
 </TD>
  <TD>
The lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument.lesson> Trigonometric functions of half argument</A> and 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument-Examples.lesson>Trigonometric functions of half argument - Examples</A>

</TD>
 </TR>
</TABLE>

<B>Miscellaneous Trigonometry problems</B>

The lesson <A HREF=http://www.algebra.com/algebra/homework/Trigonometry-basics/Miscellaneous-Trigonometry-problems.lesson>Miscellaneous Trigonometry problems</A>

Lesson Addition and subtraction of trigonometric functions - Examples

Source code of 'Addition and subtraction of trigonometric functions - Examples'