Lesson Trigonometric functions of half argument

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<H2>Trigonometric functions of half argument - Examples</H2>

The formulas for the Trigonometric functions of half argument are:

{{{sin^2(alpha/2) = (1-cos(alpha))/2}}}, &nbsp;&nbsp;{{{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))}}}, &nbsp;&nbsp;{{{cos(alpha) = (1-tan^2(alpha/2))/(1+tan^2(alpha/2))}}}, &nbsp;&nbsp;{{{tan(alpha) = 2*tan(alpha/2)/(1-tan^2(alpha/2))}}}.

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

<H3>Problem 1</H3>Find &nbsp;sin(22°30'), &nbsp;sin(22°30'), &nbsp;tan(22°30').

<B>Solution</B>

First, &nbsp;calculate &nbsp;sin(22°30').
Since &nbsp;<B>22°30'</B> = 45°/2={{{pi/8}}}, &nbsp;you can apply the formula of half argument for sines &nbsp;(see the lesson &nbsp;<A HREF= http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument.lesson>Trigonometric functions of half argument</A>&nbsp; in this module):
sin(22°30') = {{{sqrt((1-cos(pi/4))/2)}}} = {{{sqrt((1-sqrt(2)/2)/2)}}} = {{{sqrt((2-sqrt(2))/4)}}} = {{{sqrt(2-sqrt(2))/2}}}.

Similarly, &nbsp;cos(22°30') = {{{sqrt((1+cos(pi/4))/2)}}} = {{{sqrt((1+sqrt(2)/2)/2)}}} = {{{sqrt((2+sqrt(2))/4)}}} = {{{sqrt(2+sqrt(2))/2}}}.

Hence,
tan(22°30') = sin(22°30')/cos(22°30') = {{{sqrt((2-sqrt(2))/(2+sqrt(2)))}}} = (further transformations) = {{{sqrt( ((2-sqrt(2))*(2-sqrt(2))) / ((2+sqrt(2))*(2-sqrt(2))) )}}} = {{{sqrt( ((2-sqrt(2))^2)/(2^2-(sqrt(2))^2) )}}} = {{{(2-sqrt(2))/sqrt(2)}}} = {{{sqrt(2)-1}}}.

<H3>Problem 2</H3>Find &nbsp;cos(15°), &nbsp;sin(15°), &nbsp;tan(15°).

<B>Solution</B>

First, &nbsp;find sin(15°). &nbsp;Use the formula of half argument for sines (see the lesson &nbsp;<A HREF= http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument.lesson>Trigonometric functions of half argument</A>&nbsp; in this module):

sin^2(15°) = (1-cos(30°)/2 = {{{(1-sqrt(3)/2)/2}}} = {{{(2-sqrt(3))/4}}}.

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

Now, &nbsp;find cos(15°). &nbsp;Use formula for cosines of half argument, &nbsp;which is the second formula in this lesson. &nbsp;You have

cos^2(15°) = (1+cos(30°)/2 = {{{(1+sqrt(3)/2)/2}}} = {{{(2+sqrt(3))/4}}}.

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

Consequently, 
tan(15°) = sin(15°)/cos(15°) = {{{sqrt(2-sqrt(3))/sqrt(2+sqrt(3))}}} = (further transformations) = {{{sqrt( ((2-sqrt(3))*(2-sqrt(3))) / ((2+sqrt(3))*(2-sqrt(3))) )}}} = {{{sqrt( ((2-sqrt(3))^2)/(2^2 - (sqrt(3))^2) )}}} = {{{ (2-sqrt(3))/ sqrt(4-3)}}} = {{{(2-sqrt(3))/ sqrt(1)}}} = {{{2-sqrt(3)}}}.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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 <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;<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;<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;<B>Trigonometric functions of half argument - Examples</B> (this lesson)

</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>

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 Trigonometric functions of half argument - Examples

Source code of 'Trigonometric functions of half argument - Examples'