Lesson Addition and subtraction of trigonometric functions

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<H2>Addition and subtraction of trigonometric functions</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))}}}

In this lesson you can learn how to prove these formulas.

<H3>Proof of the sines addition formula</H3>

We are going to prove the formula

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

The proof is very simple and straightforward.

Introduce two new angles by formulas

{{{gamma = (alpha + beta)/2}}}, {{{delta = (alpha - beta)/2}}}.

Note that

{{{gamma + delta = (alpha + beta)/2 + (alpha - beta)/2 = alpha}}},

{{{gamma - delta = (alpha + beta)/2 - (alpha - beta)/2 = beta}}}.

Now, &nbsp;you can express &nbsp;{{{sin(alpha)}}}&nbsp; as &nbsp;{{{sin(gamma + delta)}}}&nbsp; using the addition formula for sines:

{{{sin(alpha) = sin(gamma + delta) = sin(gamma)*cos(delta) + cos(gamma)*sin(delta)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(this addition formula was proved in the lesson &nbsp;<A HREF= http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A>&nbsp; in this module)

and express &nbsp;{{{sin(beta)}}}&nbsp; as &nbsp;{{{sin(gamma - delta)}}}&nbsp; using the subtraction formula for sines:

{{{sin(beta) = sin(gamma - delta) = sin(gamma)*cos(delta) - cos(gamma)*sin(delta)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(this subtraction formula was proved in the lesson &nbsp;<A HREF= http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A>&nbsp; in this module).

Add left sides and right sides of these two formulas. &nbsp;You get

{{{sin(alpha) + sin(beta) = 2*sin(gamma)*cos(delta)}}},

what is exactly what we were going to prove.

<H3>Proof of the sines subtraction formula</H3>

We are going to prove the formula

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

The proof is very similar to the previous one.

Use the same angles &nbsp;{{{gamma}}}&nbsp; and &nbsp;{{{delta}}}&nbsp; as before. &nbsp;Express &nbsp;{{{sin(alpha)}}}&nbsp; as &nbsp;{{{sin(gamma + delta)}}}&nbsp; using the same addition formula for sines:

{{{sin(alpha) = sin(gamma + delta) = sin(gamma)*cos(delta) + cos(gamma)*sin(delta)}}}

and express &nbsp;{{{sin(beta)}}}&nbsp; as &nbsp;{{{sin(gamma - delta)}}}&nbsp; using the same subtraction formula for sines:

{{{sin(beta) = sin(gamma - delta) = sin(gamma)*cos(delta) - cos(gamma)*sin(delta)}}}.

Now, &nbsp;subtract the second formula from the first one side by side. &nbsp;You get

{{{sin(alpha) - sin(beta) = 2*cos(gamma)*sin(delta)}}},

which is exactly what we were going to prove.

Proofs of the addition and subtraction formulas for cosines are very similar to that of sines.

<H3>Proof of the tangents addition formula</H3>

We are going to prove the formula

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

Start from the left side and transform it step by step:

{{{sin(alpha)/cos(alpha) + sin(beta)/cos(beta) = (sin(alpha)*cos(beta)+cos(alpha)*sin(beta))/(cos(alpha)*cos(beta))}}} &nbsp;&nbsp;(after converting the fractions to the common denominator)

= {{{sin(alpha+beta)/(cos(alpha)*cos(beta))}}} &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;(after using the formula of sines for the sum of angles)

This is exactly what we were going to prove.

<H3>Proof of the tangents subtraction formula</H3>

The proof is similar to the previous one.

For examples of applications of these formulas see the lesson &nbsp;<A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-of-trigonometric-functions-Examples.lesson>Addition and subtraction of trigonometric functions - Examples</A>&nbsp; in this module.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For your convenience, &nbsp;below is the list of my lessons on Trigonometry in this site in the logical order. 
They all are under the current topic &nbsp;<B>Trigonometry</B>&nbsp; in the section &nbsp;<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 &nbsp;<A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A>&nbsp; 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;<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 &nbsp;<B>Addition and subtraction of trigonometric functions</B>&nbsp; (this lesson) &nbsp;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;<A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-of-trigonometric-functions-Examples.lesson>Addition and subtraction of trigonometric functions - Examples</A>

</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;&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 &nbsp;<A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Powers-of-trigonometric-functions.lesson>Powers of Trigonometric functions</A>&nbsp; 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;<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 &nbsp;<A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-multiply-argument.lesson>Trigonometric functions of multiply argument</A>&nbsp; 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;&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 &nbsp;<A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument.lesson>Trigonometric functions of half argument</A>&nbsp; and 
&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-half-argument-Examples.lesson>Trigonometric functions of half argument - Examples</A>

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

<B>Miscellaneous Trigonometry problems</B>

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

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-II.