Lesson Miscellaneous Trigonometry problems
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<H2>Miscellaneous Trigonometry problems</H2> <H3>Problem 1</H3>Find sin(22°30'), cos(22°30'), tan(22°30'). <B>Solution</B> First, calculate sin(22°30'). Since <B>22°30'</B> = 45°/2={{{pi/8}}}, you can apply the formula of half argument for sines (see the lesson <A HREF= http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument.lesson>Trigonometric functions of half argument</A> 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, 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)))}}} = ( <B><U>simplify</U></B> ) = {{{sqrt( (2-sqrt(2))^2/( (2+sqrt(2))*(2-sqrt(2)) ) )}}} = {{{(2 -sqrt(2))/sqrt(4-2) )}}} = {{{(2-sqrt(2))/sqrt(2) )}}} = {{{sqrt(2)-1}}}. <H3>Problem 2</H3>Prove that {{{alpha + beta = pi/4}}}, if {{{alpha}}} and {{{beta}}} are acute angles and {{{tan(alpha) = 1/2}}}, {{{tan(beta) = 1/3}}}. <B>Solution</B> Calculate {{{tan(alpha+beta)}}} using the addition formula for tangents (see 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): {{{tan(alpha+beta) = (tan(alpha)+tan(beta))/(1-tan(alpha)*tan(beta)) = (1/2+1/3)/(1-1/2*1/3) = (3/6+2/6)/(1-1/6)=(5/6)/(5/6) = 1}}}. Since the angles {{{alpha}}} and {{{beta}}} are acute and {{{tan(alpha+beta) = 1}}}, we have {{{alpha + beta = pi/4}}}. <H3>Problem 3</H3>Prove that {{{alpha + beta + gamma = pi/4}}}, if {{{alpha}}}, {{{beta}}} and {{{gamma}}} are acute angles and {{{tan(alpha) = 1/2}}}, {{{tan(beta) = 1/5}}} and {{{tan(gamma) = 1/8}}}. <B>Solution</B> First, calculate {{{tan(alpha+beta)}}} using the addition formula for tangents (see 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): {{{tan(alpha+beta) = (tan(alpha)+tan(beta))/(1-tan(alpha)*tan(beta)) = (1/2+1/5)/(1-1/2*1/5) = (5/10+2/10)/(1-1/10)=(7/10)/(9/10) = 7/9}}}. Now, calculate {{{tan(alpha+beta+gamma)}}} using the same addition formula for tangents: {{{tan(alpha+beta+gamma) = (tan(alpha+beta)+tan(gamma))/(1-tan(alpha+beta)*tan(gamma)) = (7/9+1/8)/(1-7/9*1/8) = (56/72+9/72)/(1-7/72)=(65/72)/(65/72) = 1}}}. Since the angles {{{alpha}}} and {{{beta}}} are acute and {{{tan(alpha+beta)}}} is positive (equal to {{{7/9}}}), the angle {{{alpha+beta}}} is acute. Since the angles {{{alpha+beta}}} and {{{gamma}}} are acute and {{{tan(alpha+beta+gamma)}}} is positive (equal to {{{1}}}), the angle {{{alpha+beta+gamma}}} is acute. Since the angle {{{alpha+beta+gamma}}} is acute and {{{tan(alpha+beta+gamma)=1}}}, we have {{{alpha + beta + gamma = pi/4}}}. <H3>Problem 4</H3>If {{{alpha+beta+gamma = pi}}}, show that {{{sin(2alpha) + sin(2beta) + sin(2gamma) = 4*sin(alpha)*sin(beta)*sin(gamma)}}}. <B>Solution</B> First, transform the sum {{{sin(2alpha) + sin(2beta)}}} as follows: {{{sin(2alpha) + sin(2beta) = 2*sin(alpha+beta)*cos(alpha-beta) = 2*sin(pi-gamma)*cos(alpha-beta) = 2*sin(gamma)*cos(alpha-beta)}}}, Next, represent {{{sin(2gamma)}}} as {{{sin(2gamma) = 2*sin(gamma)*cos(gamma)}}}. Now, calculate {{{sin(2alpha) + sin(2beta) + sin(2gamma)}}} {{{sin(2alpha) + sin(2beta) + sin(2gamma) = 2*sin(gamma)* ( cos(alpha-beta) + cos(gamma) ) }}} = = {{{2*sin(gamma)* ( 2*cos((alpha-beta+gamma)/2)*cos((alpha-beta-gamma)/2) ) }}} = = {{{4*sin(gamma)* cos(pi/2-beta)*cos(alpha-pi/2) = 4*sin(gamma)* sin(beta)*sin(alpha)}}}. You got what you were going to prove. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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 <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Powers-of-trigonometric-functions.lesson>Powers of Trigonometric functions</A> 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 <B>Miscellaneous Trigonometry problems</B> (this lesson) 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.
