Lesson Truly elegant solution to one trigonometric equation
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<H2>Truly elegant solution to one trigonometric equation</H2> <H3>Problem 1</H3>Solve an equation {{{cos(theta)}}} - {{{sqrt(3)*sin(theta)}}} = 1. <B>Solution</B> I will show you <U>TRULY elegant</U> method of solution . . . <pre> {{{cos(theta)}}} - {{{sqrt(3)*sin(theta)}}} = 1. Multiply both sides by {{{1/2}}}. {{{(1/2)*cos(theta)}}} - {{{(sqrt(3)/2)*sin(theta)}}} = {{{1/2}}}. It can be written in this form {{{sin(pi/6)*cos(theta)}}} - {{{cos(pi/6)*sin(theta)}}} = {{{sin(pi/3)}}}. Use the formula of adding arguments for sine {{{sin((pi/6) - theta)}}} = {{{sin(pi/6)}}}. It may happen in one of the two cases <U>Case 1</U>. {{{(pi/6) - theta}}} = {{{pi/6}}}. which implies {{{theta}}} = 0, or <U>Case 2</U>. {{{(pi/6) - theta}}} = {{{pi}}} - {{{pi/6}}}, which implies {{{theta}}} = {{{(2pi)/6 - pi}}} = {{{pi/3 - pi}}} = -{{{(2pi)/3}}}. Since we consider the angles in the interval [0,2pi), it is equivalent to {{{theta}}} = {{{(4pi)/3}}}. So, we get the <U>ANSWER</U> : The given equation has two solutions in the interval [0,2pi) a) {{{theta}}} = 0, and b) {{{theta}}} = {{{(4pi)/3}}}. </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/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/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.
