Lesson Math OLYMPIAD level problems on Trigonometry
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<H2>Math OLYMPIAD level problems on Trigonometry</H2> <H3>Problem 1</H3>Let f(x) = {{{(x - sqrt(3))/(x*sqrt(3) + 1)}}}. What is f^{2012}(x), where the function is being applied/composed 2012 times? This notation indicates repeated composition of functions, not exponentiation of functions. For example, f^2(x) = f(f(x)) and not f(x)*f(x). Similarly, f^3(x) = f(f(f(x))). <B>Solution</B> This problem allows absolutely unexpected and elegant solution. <pre> Let x = tan(a) (which means "let a = arctan(x)"). Then f(x) = {{{(tan(a) - sqrt(3))/(tan(a)*sqrt(3)+1)}}} = {{{(tan(a) - tan(pi/3))/(1+tan(a)*tan(pi/3))}}} = {{{tan(a-pi/3)}}}. It means that taking f(x) for x= tan(a) returns {{{tan(a-pi/3)}}}. Obviously, that taking the composition (fof)(x) = f(f(x)) will return {{{tan(a-2pi/3)}}}; taking the composition (fofof)(x) = f(f(f(x))) will return {{{tan(a-3pi/3)}}}; . . . . and so on . . . taking the composition f^{2012}(x) = f(f(f...f(x)))...) will return {{{tan(a-2012pi/3)}}}. It easy to calculate: {{{2012pi/3}}} = {{{670*pi}}} + {{{2pi/3}}}; THEREFORE {{{tan(a-2012pi/3)}}} = {{{tan(a-670pi - 2pi/3)}}} = {{{tan(a-2pi/3)}}} = {{{(tan(a) - tan(2pi/3))/(1 + tan(a)*tan(2pi/3))}}} = {{{(tan(a)-(-sqrt(3)))/(1+tan(a)*(-sqrt(3)))}}} = {{{(x+sqrt(3))/(1-sqrt(3)*x)}}}. <U>ANSWER</U> <U>ANSWER</U>. f^{2012}(x) = {{{(x+sqrt(3))/(1-sqrt(3)*x)}}}. </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/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/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/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.
