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 <A HREF= http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A> in this module: {{{cos(alpha + beta) = cos(alpha)*cos(beta) - sin(alpha)*sin(beta)}}}. Simply take {{{beta = alpha}}} in this formula. You get {{{cos(2alpha) = cos^2(alpha) - sin^2(alpha)}}}. Now, substitute {{{sin^2(alpha) = 1 - cos^2(alpha)}}}. to the previous equation. You get {{{cos(2alpha) = cos^2(alpha) - (1 - cos^2(alpha)) = 2*cos^2(alpha) - 1}}}. Make simple rearrangements in the line above, 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. 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, 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 <A HREF= http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A> of this module in the form {{{cos(3alpha) = cos(2alpha + alpha) = cos(2alpha)*cos(alpha) - sin(2alpha)*sin(alpha)}}}. (*) For {{{cos(2alpha)}}} we just have ready the expressions {{{cos(2alpha) = 2*cos^2(alpha) - 1}}}, which was proved above. For {{{sin(2alpha)}}} 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, substitute these expressions for {{{cos(2alpha)}}} and {{{sin(2alpha)}}} to the formula (*) above. You get {{{cos(3alpha) = (2*cos^2(alpha) - 1)*cos(alpha) - 2*sin(alpha)*cos(alpha)*sin(alpha)}}} ={{{(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)}}} ={{{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, and you get exactly what we are going to prove. The proof is completed. For examples of applications of these formulas see the lesson <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Powers-of=trigonometric-functions-Examples.lesson>Powers of trigonometric functions - Examples</A> in this module. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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> The lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A> and <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> 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 <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> The lessons <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Product-of-trigonometric-functions.lesson>Product of trigonometric functions</A> and <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> The lessons <B>Powers of Trigonometric functions</B> (this lesson) and <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> 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 <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 <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> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.