Lesson Solving more complicated problems on trigonometric equations
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<H2>Solving more complicated problems on trigonometric equations</H2> In this lesson you will find the solutions of these trigonometric equations: 1. {{{2cos^2(x) + cos(x)}}} = {{{0}}}. 2. {{{2sin^2(x) -7sin(x) + 3}}} = {{{0}}}. 3. {{{2cos^2(theta)}}} = {{{sin(theta)+1}}}. 4. {{{3cos(beta) + 3}}} = {{{2sin^2(beta)}}}. 5. {{{cos(2x) + 6sin^2(x)}}} = {{{4}}}. <H3>Problem 1</H3>Solve an equation {{{2cos^2(x) + cos(x)}}} = {{{0}}}. <B>Solution</B> <pre> {{{2cos^2(x) + cos(x)}}} = {{{0}}}. (1) Factor the left side. You will get cos(x)*(2cos(x) + 1) = 0. The last equation splits in two separate equations: 1. cos(x) = 0 ---> x = {{{pi/2}}} and/or x = {{{-pi/2}}}. In the interval [{{{0}}},{{{2pi}}}) there are no other solutions. If you want to write general solution then x = {{{k*pi}}}, where k is any integer, k = 0, +/-1, +/-2 . . . 2. 2cos(x) + 1 = 0 ---> cos(x) = {{{-1/2}}} ---> x = {{{2pi/3}}} and/or x = {{{4pi/3}}}. In the interval [{{{0}}},{{{2pi}}}) there are no other solutions. If you want to write a general solution, then x = +/- {{{pi/3}}} + {{{(2k+1)*pi}}}, where k is any integer, k = 0, +/-1, +/-2 . . . <B>Answer</B>. The solution of the original equation (1) is the union of the solutions to n.1 and n.2. </pre> <H3>Problem 2</H3>Solve an equation {{{2sin^2(x) - 7sin x + 3}}} = {{{0}}} for all real values of x. <B>Solution</B> <pre> {{{2sin^2(x) - 7 sin x + 3}}} = {{{0}}}. Group the terms to get factoring: {{{2sin^2(x) - 6sin(x) - sin(x) + 3}}} = {{{0}}}. {{{2sin(x)*(sin(x)-3) - (sin(x)-3)}}} = {{{0}}}, (sin(x)-3)*(2sin(x)-1) = 0. The last equation splits in two equations: 1. sin(x) = 3 (this equation has no solutions) or 2. sin(x) = 1/2 ---> x = {{{pi/6 + 2k*pi}}} or x = {{{(5/6)pi + 2k*pi}}}, where k is any integer k = 0, +/-1, +/-2, . . . <B>Answer</B>. The solutions are x = {{{pi/6 + 2k*pi}}} or x = {{{(5/6)pi + 2k*pi}}}, where k is any integer. </pre> <H3>Problem 3</H3>Solve an equation {{{2cos^2(theta)}}} = {{{sin(theta)+1}}}. <B>Solution</B> <pre> {{{2cos^2(theta)}}} = {{{sin(theta)+1}}}. Replace {{{cos^2(theta)}}} by {{{1 - sin^2(theta)}}} to get the equation for sine only. You will get {{{2*(1-sin^2(theta))}}} = {{{sin(theta)+1}}}, or {{{2 - 2sin^2(theta)}}} = {{{sin(theta)+1}}}, or {{{2*sin^2(theta) + sin(theta) - 1}}} = {{{0}}}, or (after factoring left side) {{{(2*sin(theta) - 1)*(sin(theta) + 1)}}} = {{{0}}}. The last equation splits in two equations: 1. {{{2*sin(theta) - 1}}} = {{{0}}} ---> {{{sin(theta)}}} = {{{1/2}}} ---> {{{theta}}} = {{{pi/6 + 2k*pi}}} or {{{theta}}} = {{{5pi/6 + 2k*pi}}}, k = 0, +/-1, +/-2, . . . 2. {{{sin(theta) + 1}}} = {{{0}}} ---> {{{sin(theta)}}} = {{{-1}}} ---> {{{theta}}} = {{{3pi/4 + 2k*pi}}} = {{{(2k+1)*pi}}}, k = 0, +/-1, +/-2, . . . <B>Answer</B>. The solution of the original equation is the union of the solutions to n.1 and n.2. </pre> <H3>Problem 4</H3>Solve an equation {{{3cos(beta) + 3}}} = {{{2sin^2(beta)}}}. <B>Solution</B> <pre> {{{3*cos(beta) + 3}}} = {{{2sin^2(beta)}}}. Replace {{{sin^2(beta)}}} by {{{1 - cos^2(beta)}}} to get the equation for cosine only. You will get {{{3*cos(beta) + 3}}} = {{{2*(1 - cos^2(beta))}}}, or {{{3*cos(beta) + 3}}} = {{{2 - 2*cos^2(beta)}}}, or {{{2*cos^2(beta) + 3*cos(beta) + 1}}} = {{{0}}}, or (after factoring left side) {{{(2*cos(beta) + 1)*(cos(beta) + 1)}}} = {{{0}}}. It splits in two equations: 1. {{{2*cos(beta) + 1}}} = {{{0}}} ---> {{{cos(beta)}}} = {{{-1/2}}} ---> {{{beta}}} = {{{2pi/3 + 2k*pi}}} or beta = {{{4pi/3 + 2k*pi}}}, k = 0, +/-1, +/-2, . . . 2. {{{cos(beta) + 1}}} = {{{0}}} ---> {{{cos(beta)}}} = {{{-1}}} ---> {{{beta}}} = {{{pi + 2k*pi}}} = {{{(2k+1)*pi}}}, k = 0, +/-1, +/-2, . . . <B>Answer</B>. The solution of the original equation is the union of the solutions to n.1 and n.2. </pre> <H3>Problem 5</H3>Solve an equation {{{cos(2x) + 6sin^2(x)}}} = {{{4}}}. <B>Solution</B> <pre> {{{cos(2x) + 6sin^2(x)}}} = {{{4}}}. The way to solve it is to transform the original equation into the equation for sine only. To do it, use the formula for cosine of doubled argument cos(2x) = {{{cos^2(x) - sin^2(x)}}} and replace {{{cos^2(x)}}} by {{{1-sin^2(x)}}}. You will get cos(2x) = {{{1 - 2*sin^2(x)}}}. Now substitute it into original equation. You will get {{{1 - 2*sin^2(x) + 6*sin^2(x)}}} = {{{4}}}. Collect like terms. You will get {{{4*sin^2(x)}}} = {{{3}}}, or {{{sin^2(x)}}} = {{{3/4}}}, or (after taking the square root from both sides) sin(x) = +/- {{{sqrt(3)/2}}}. The solutions are x = {{{pi/3}}}, {{{2pi/3}}}, {{{4pi/3}}}, {{{5pi/3}}}. In the interval [{{{0}}},{{{2pi}}}) these are all solutions. If you want to write a general solution, then x = +/- {{{pi/3}}} + {{{k*pi}}}, where k is any integer, k = 0, +/-1, +/-2 . . . </pre> 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-typical-problems-on-trigonometric-equations.lesson>Solving typical 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 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.
