Lesson Powers of trigonometric functions

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<H2>Powers of trigonometric functions - Examples</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)}}}.

The proofs of these formulas are presented in the lesson &nbsp;<A HREF= http://www.algebra.com/algebra/homework/Trigonometry-basics/Powers-of-trigonometric-functions.lesson>Powers of trigonometric functions</A>&nbsp; in this module.
Below are examples of applications of these formulas.

<H3>Example 1</H3>Find &nbsp;sin(15°), &nbsp;cos(15°), &nbsp;tan(15°).

<B>Solution</B>

First, &nbsp;find &nbsp;sin(15°). 
Put &nbsp;{{{alpha}}}&nbsp; = 15°. &nbsp;Note that &nbsp;{{{2alpha}}} = 30° &nbsp;and use the formula for square of sines:

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

Substitute &nbsp;{{{alpha}}} = 15°, &nbsp;{{{2alpha}}} = 30° &nbsp;and &nbsp;{{{cos(2alpha)}}} = cos(30°) = {{{sqrt(3)/2}}} &nbsp;into this formula.&nbsp; You get the equation
sin^2(15°) = {{{-(1/2)*(sqrt(3)/2) + 1/2}}}, &nbsp;or

sin^2(15°) = {{{(2-sqrt(3))/4}}}.

Hence,
sin(15°) = {{{sqrt(2-sqrt(3))/2}}}.

Having calculated &nbsp;sin(15°), &nbsp;you can easily calculate &nbsp;cos(15°):
cos^2(15°) = 1 - sin^2(15°) = {{{1 - (2-sqrt(3))/4 = (2+sqrt(3))/4}}},

hence,
cos(15°) = {{{sqrt(2+sqrt(3))/2}}}.

Now, 
tan(15°) =  sin(15°)/cos(15°) = {{{sqrt(2-sqrt(3))/sqrt(2+sqrt(3))}}}.

Note that &nbsp;sin(15°), &nbsp;cos(15°)&nbsp; and &nbsp;tan(15°)&nbsp; were just calculated by other ways in lessons 
&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>&nbsp; and 
&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>&nbsp; in this module.
Please make sure that all relevant results from these lessons are identical.

<H3>Example 2</H3>Find &nbsp;sin(18°).

<B>Solution</B>

Let us denote &nbsp;{{{alpha}}} = 18°.
Then &nbsp;{{{5alpha}}} = 90°, 
hence, &nbsp;{{{2alpha}}} = 90°-{{{3alpha}}}.

Therefore, 
{{{sin(2alpha)}}} = {{{sin(pi/2-3alpha)}}}, 
and consequently
{{{sin(2alpha) = cos(3alpha)}}} 
(which is, &nbsp;actually, &nbsp;the obvious equality &nbsp;sin(36°) = cos(54°)).

Now, &nbsp;apply the formula for the double argument to sines at the left side and the formula for the triple argument to cosines at the right side.

<BLOCKQUOTE>The formula for the double argument to sines follows from the addition formula for sines: 
{{{sin(2alpha) = sin(alpha + alpha) = sin(alpha)*cos(alpha) + cos(alpha)*sin(alpha) = 2*sin(alpha)*cos(alpha)}}}.

The formula for the triple argument to sines follows from the third formula of this lesson:
{{{cos(3alpha) = 4*cos^3(alpha) - 3*cos(alpha)}}}.
</BLOCKQUOTE>

After applying these formulas you get
{{{2*sin(alpha)*cos(alpha) = 4*cos^3(alpha) - 3*cos(alpha)}}}.

Since &nbsp;{{{cos(alpha)}}}&nbsp; is not equal to zero, &nbsp;you can divide both sides of the last equality by &nbsp;{{{cos(alpha)}}}. &nbsp;You get the equation
{{{2*sin(alpha) = 4*cos^2(alpha) - 3}}}.

Now, &nbsp;introduce {{{x=sin(alpha)}}}&nbsp; for short and replace &nbsp;{{{cos^2(alpha) = 1-x^2}}}&nbsp; in the preceding formula. &nbsp;You get the equation
{{{2x = 4(1-x^2) -3}}},

or, &nbsp;after simplifying,
{{{4x^2 + 2x - 1 = 0}}}.

This is the quadratic equation. &nbsp;Solve it using the &nbsp;<B>quadratic formula</B>&nbsp; (see the lesson &nbsp;<A HREF = http://www.algebra.com/algebra/homework/quadratic/lessons/Introduction-Into-Quadratics.lesson>Introduction into Quadratic Equations</A>&nbsp; in this site).
You get two roots

{{{x[1] = (-2 + sqrt(4 + 4*4*1))/(2*4) = (-2 + sqrt(20))/8 = (-1+sqrt(5))/4}}}, &nbsp;and
{{{x[2] = (-2 - sqrt(4 + 4*4*1))/(2*4) = (-2 - sqrt(20))/8 = (-1-sqrt(5))/4}}}.

Only the first root fits &nbsp;(the second root doesn't fit because it is negative).
So, &nbsp;the &nbsp;<B>answer</B>&nbsp; is: &nbsp;sin(18°) = {{{(-1+sqrt(5))/4}}}.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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;<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;<B>Powers of Trigonometric functions - Examples</B>&nbsp; (this lesson)

</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.

Lesson Powers of trigonometric functions - Examples

Source code of 'Powers of trigonometric functions - Examples'