Lesson Solving typical problems on trigonometric equations
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<H2>Solving typical problems on trigonometric equations</H2> In this lesson you will find the solutions of these typical trigonometric equations: 1. {{{8cos^2(x)}}} = {{{2}}}. 2. {{{4sin^2(x) -3}}} = {{{0}}}. 3. {{{tan(x) + sqrt(3)}}} = {{{sec(x)}}}. 4. {{{sin(x) + cos(x)}}} = {{{-1}}}. 5. {{{sin(x) - cos(x)}}} = {{{sqrt(2)}}}. 6. {{{sqrt(3)*cos(x) + sin(x)}}} = {{{1}}}. <H3>Problem 1</H3>Solve an equation {{{8cos^2(x)}}} = {{{2}}} in the interval [{{{0}}},{{{2pi}}}) <B>Solution</B> <pre> {{{8cos^2(x)}}} = {{{2}}} ---> cos^2(x) = {{{2/8}}} = {{{1/4}}}, cos(x) = +/-{{{sqrt(1/4)}}} = +/-{{{1/2}}}. cos(x) = {{{1/2}}} ---> x = {{{pi/3}}} or/and x = {{{(5pi)/3}}}. cos(x) = {{{-1/2}}} ---> x = {{{(2pi)/3}}} or/and x = {{{(4pi)/3}}}. <U>Answer</U>. x = {{{pi/3}}}, {{{(2pi)/3}}}, {{{(4pi)/3}}} and {{{(5pi)/3}}}. </pre> <H3>Problem 2</H3>Solve an equation {{{4sin^2(x) -3}}} = {{{0}}} in the following domain 0 <= x < {{{2pi}}}. <B>Solution</B> </pre> <H3>Problem 3</H3>Solve an equation {{{tan(x) + sqrt(3)}}} = {{{sec(x)}}}. <B>Solution</B> <pre> {{{tan(x) + sqrt(3)}}} = sec(x) is the same as {{{sin(x)/cos(x)}}} + {{{sqrt(3)}}} = {{{1/cos(x)}}}. Multiply both sides by cos(x). You will get {{{sin(x)}}} + {{{sqrt(3)*cos(x)}}} = 1. Multiply both sides by {{{1/2}}}. You will get {{{(1/2)*sin(x)}}} + {{{(sqrt(3)/2)*cos(x)}}} = {{{1/2}}}. Recall that {{{1/2}}} = {{{cos(pi/3)}}}, {{{sqrt(3)/2}}} = {{{sin(pi/3)}}}. Therefore, you can write the last equation as {{{cos(pi/3)*sin(x) + sin(pi/3)*cos(x)}}} = {{{1/2}}}. Apply the addition formula for sine. ( It is cos(a)*sin(b) + sin(a)*cos(b) = sin(a+b). See the lesson <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A> in this site ). You will get {{{sin(x + pi/3)}}} = {{{1/2}}}. It implies {{{x + pi/3}}} = {{{pi/6}}} or {{{x + pi/3}}} = {{{5pi/6}}}. Hence, x = {{{pi/6-pi/3}}} = {{{-pi/6}}} or x = {{{5pi/6-pi/3}}} = {{{3pi/6}}} = {{{pi/2}}}. The last root doesn't fit due to "sec" in the original equation. <U>Answer</U>. x = {{{-pi/6}}}, or {{{-pi/6 + 2k*pi}}} for any integer "k". </pre> <H3>Problem 4</H3>Solve for x: {{{sin(x) + cos(x)}}} = {{{-1}}}. <B>Solution</B> <pre> sin(x) + cos(x) = -1. (1) (It is the original equation) Square its both sides. You will get {{{sin^2(x) + 2*sin(x)*cos(x) + cos^2(x)}}} = {{{1}}}. (2) From the other side, there is an identity {{{sin^2(x) + cos^2(x)}}} == {{{1}}}. (3) Comparing (2) and (3), you get 2*sin(x)*cos(x) = 0, or sin(x)*cos(x) = 0. (4) Equation (4) splits in two independent equations 1) sin(x) = 0 ---> x = {{{k*pi}}}, k = 0. +/-1. +/-2, . . . (5) 2) cos(x) = 0 ---> x = {{{pi/2 + k*pi}}}, k = 0. +/-1. +/-2, . . . (6) Now we should check which of the found values (5), (6) satisfy the original equation. Of the set (5), all x satisfy sin(x) = 0. Hence, only those of (5) satisfy the original equation where cos(x) = -1. They are x = {{{pi + 2n*pi}}}, n = 0, +/-1. +/-2, . . . , or x = {{{(2n+1)*pi}}}, n = 0, +/-1. +/-2, . . . , (5'). Of the set (6), all x satisfy cos(x) = 0. Hence, only those of (6) satisfy the original equation where sin(x) = -1. They are x = {{{3pi/2 + 2n*pi}}}, n = 0, +/-1. +/-2, . . . , or x = {{{(3/2 + 2n)*pi}}}, n = 0, +/-1. +/-2, . . . , (6'). <U>Answer</U>. The union of the sets (5') and (6') is the solution of the original equation. </pre> <H3>Problem 5</H3>Solve an equation {{{sin(x) - cos(x)}}} = {{{sqrt(2)}}}. <B>Solution</B> <pre> {{{sin(x) - cos(x)}}} = {{{sqrt(2)}}}. Multiply both sides by {{{sqrt(2)/2}}}. You will get {{{(sqrt(2)/2)*sin(x) - (sqrt(2)/2)*cos(x)}}} = 1. (1) Notice and use that {{{sqrt(2)/2}}} = {{{cos(pi/4)}}} = {{{sin(pi/4)}}}. Then from (1) you will get {{{cos(pi/4)*sin(x) - sin(pi/4)*cos(x)}}} = 1. (2) Now use the formula sin(a)*cos(b) - cos(a)*sin(b) = sin(a-b). Then from (2) you will get {{{-sin(pi/4-theta)}}} = 1. It implies {{{pi/4 - x}}} = {{{3pi/2}}}. Then {{{x}}} = {{{pi/4 - 3pi/2}}} = {{{-5pi/4}}}. It is the same as {{{x}}} = {{{3pi/4}}} in the interval 0 <= {{{theta}}} < {{{2pi}}}. </pre> The plot below confirms the solution ( {{{3pi/4}}} ~= 2.33 ) {{{graph( 330, 330, -0.5, 6.5, -2.5, 2.5, sin(x) - cos(x), sqrt(2) )}}} Plots y = {{{sin(x) - cos(x)}}} and y = {{{sqrt(2)}}} <H3>Problem 6</H3>Find the general solution to an equation {{{sqrt(3)*cos(x) + sin(x)}}} = {{{1}}}. <B>Solution</B> <pre> {{{sqrt(3)*cos(x) + sin(x)}}} = {{{1}}}. Multiply both sides by {{{1/2}}}. You will get {{{(sqrt(3)/2)*cos(x) + (1/2)*sin(x)}}} = {{{1/2}}}. (1) Notice that {{{sqrt(3)/2}}} = {{{sin(pi/3)}}}, {{{1/2}}} = {{{cos(pi/3)}}}. Substitute it into the left side of (1). You will get {{{sin(pi/3)*cos(x) + cos(pi/3)*sin(x)}}} = {{{1/2}}}. (2) Apply the formula sin(a)*cos(b) + cos(a)*sin(b) = sin(a+b) to the left side of (2). You ill get {{{sin(pi/3 + x)}}} = {{{1/2}}}. (3) It implies {{{pi/3 + x}}} = {{{pi/6 + 2k*pi}}}, k = 0, +/-1, +/-2, . . . or {{{pi/3 + x}}} = {{{5pi/6 + 2k*pi}}}, k = 0, +/-1, +/-2, . . . Thus there are two sets of solutions: 1. x = {{{pi/6 - pi/3 + 2k*pi}}} = {{{-pi/6 + 2k*pi}}}, which is equivalent to x = {{{11pi/6 + 2k*pi}}}, and the other family 2. x = {{{5pi/6 - pi/3 + 2k*pi}}} = {{{pi/2 + 2k*pi}}} <U>Answer</U>. There are two sets of solutions: 1) x = {{{11pi/6 + 2k*pi}}} and 2) x = {{{pi/2 + 2k*pi}}}, k = 0, +/-1, +/-2, . . . </pre> The plot below confirms these solutions. {{{graph( 330, 330, -0.5, 6.5, -2.5, 2.5, sqrt(3)*cos(x) + sin(x), 1 )}}} Plots y = {{{sqrt(3)*cos(x) + sin(x)}}} and y = 1 My other lessons on calculating trig functions and solving trig equations in this site are - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Calculating-trigonometric-functions-of-angles.lesson>Calculating trigonometric functions of angles</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Selected-problems-from-the-archive-on-calculating-trig-functions-of-angles.lesson>Advanced problems on calculating trigonometric functions of angles</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Evaluating-trigonometric-expressions.lesson>Evaluating trigonometric expressions</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solve-these-trigonometry-problems-without-using-a-calculator.lesson>Solve these trigonometry problems without using a calculator</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Finding-the-slope-of-the-bisector-to-the-angle-formed-by-two-given-lines-in-a-coordinate-plane.lesson>Finding the slope of the bisector to the angle formed by two given lines in a coordinate plane</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-simple-problems-on-trigonometric-equations.lesson>Solving simple problems on trigonometric equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-more-complicated-problems-on-trigonometric-equations.lesson>Solving more complicated problems on trigonometric equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solved-problems-on-trigonometric-equations.lesson>Solving advanced problems on trigonometric equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Challenging-problems-on-trigonometric-equations.lesson>Challenging problems on trigonometric equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Miscellaneous-problems-on-solving-trigonometric-equations.lesson>Miscellaneous problems on solving trigonometric equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-twisted-trigonometry-equations.lesson>Solving twisted trigonometric equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Truly-elegant-solution-to-one-trigonometric-equation.lesson>Truly elegant solution to one trigonometric equation</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Non-standard-Trigonometry-problems.lesson>Non-standard Trigonometry problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Proving-Trigonometry-identities.lesson>Proving Trigonometry identities</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Calculating-1sin%281%B0%29%2B2sin%282%B0%29%2B3sin%283%B0%29%2B-%2B180sin%28180%B0%29-.lesson>Calculating the sum 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°)</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Find-the-height.lesson>Find the height</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Word-problems-on-Trigonometric-functions.lesson>Word problems on Trigonometric functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-upper-league-Trigonometry-equations.lesson>Solving upper-league Trigonometry equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Math-OLYMPIAD-level-problems-on-Trigonometry.lesson>Math OLYMPIAD level problems on Trigonometry</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-entertainment-problems.lesson>Trigonometry entertainment problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/OVERVIEW-of-lessons-on-calculating-trig-functions-and-solving-trig-equations.lesson>OVERVIEW of lessons on calculating trig functions and solving trig equations</A> Use this file/link <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.
