Lesson Powers of trigonometric functions

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<H2>Powers of trigonometric functions</H2>

The formulas for Powers of trigonometric functions are:

{{{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)}}}.

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

<H3>Proof of the cosines square formula</H3>

We are going to prove the formula

{{{cos^2(alpha) = (1/2)*cos(2alpha) + 1/2}}}.

The proof is very simple and straightforward.

It is based on the addition formula for cosines of 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:

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

Simply take &nbsp;{{{beta = alpha}}}&nbsp; in this formula. &nbsp;You get

{{{cos(2alpha) = cos^2(alpha) - sin^2(alpha)}}}.

Now, &nbsp;substitute 
{{{sin^2(alpha) = 1 - cos^2(alpha)}}}.

to the previous equation. &nbsp;You get
{{{cos(2alpha) = cos^2(alpha) - (1 - cos^2(alpha)) = 2*cos^2(alpha) - 1}}}.

Make simple rearrangements in the line above, &nbsp;and you get exactly what we are going to prove. 
The proof is completed.

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

We are going to prove the formula

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

Start from 
{{{sin^2(alpha) = 1 - cos^2(alpha)}}},

which is kind of the basic formulas.
Substitute
{{{cos^2(alpha) = (1/2)*cos(2alpha) + 1/2}}},

the formula which was proved above. &nbsp;You get
{{{sin^2(alpha) = 1 - ((1/2)*cos(2alpha) + 1/2) = 1/2 - (1/2)*cos(2alpha)}}}.

Make simple rearrangements in the line above, &nbsp;and you get exactly what we are going to prove. 
The proof is completed.

<H3>Proof of the cosines cube formula</H3>

We are going to prove the formula

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

Let us apply the addition formula for cosines of the lesson &nbsp;<A HREF= http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A>&nbsp; of this module
in the form

{{{cos(3alpha) = cos(2alpha + alpha) = cos(2alpha)*cos(alpha) - sin(2alpha)*sin(alpha)}}}. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*)

For &nbsp;{{{cos(2alpha)}}}&nbsp; we just have ready the expressions 
{{{cos(2alpha) = 2*cos^2(alpha) - 1}}}, 
which was proved above.

For &nbsp;{{{sin(2alpha)}}}&nbsp; the general addition formula for sines gives
{{{sin(2alpha) = sin(alpha + alpha) = sin(alpha)*cos(alpha)+cos(alpha)*sin(alpha) = 2*sin(alpha)*cos(alpha)}}}.

Now, &nbsp;substitute these expressions for &nbsp;{{{cos(2alpha)}}}&nbsp; and &nbsp;{{{sin(2alpha)}}}&nbsp; to the formula &nbsp;(*)&nbsp; above. &nbsp;You get
{{{cos(3alpha) = (2*cos^2(alpha) - 1)*cos(alpha) - 2*sin(alpha)*cos(alpha)*sin(alpha)}}} 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;={{{(2*cos^2(alpha) - 1)*cos(alpha) - 2*sin^2(alpha)*cos(alpha) = (2*cos^2(alpha) - 1)*cos(alpha) - 2*(1-cos^2(alpha))*cos(alpha)}}} 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;={{{2*cos^3(alpha) - cos(alpha) + 2*cos^3(alpha) -2*cos(alpha) = 4*cos^3(alpha) - 3*cos(alpha)}}}.

Make the last rearrangements in the lines above, &nbsp;and you get exactly what we are going to prove. 
The proof is completed.

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