Lesson Solving equation by analyzing function
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<H2>Solving equation by analyzing function</H2> <H3>Problem 1</H3>Solve an equation {{{sqrt(sin(x) - sqrt(sin(x) + cos(x)))}}} = {{{cos(x)}}}. <B>Solution</B> <pre> Our starting equation is {{{sqrt(sin(x) - sqrt(sin(x) + cos(x)))}}} = cos(x). (1) Let's consider function {{{sin(x) - sqrt(sin(x) + cos(x))}}} first, which is under the outer square root of the left side. Inside the first quadrant, QI, when 0 < x < {{{pi/2}}}, both sin(x) and cos(x) are positive. Therefore, the sum sin(x) + cos(x) is greater than sin(x): sin(x) + cos(x) > sin(x). (2) Square root is a monotonic function of its argument; therefore, from inequality (1) we have {{{sqrt(sin(x) + cos(x))}}} > {{{sqrt(sin(x))}}}. (3) Since inside the first quadrant, QI, sin(x) is less than 1, we have {{{sqrt(sin(x))}}} > sin(x). (4) So, combining (3) and (4), we have sin(x) < {{{sqrt(sin(x) + cos(x))}}}. It means that inside QI the expression under the outer square root is negative; so, left side of the given equation is not defined. Let's inspect endpoints of QI, angles x = 0 and x = {{{pi/2}}}. At x = 0, left side of the given equation is not defined. At x = {{{pi/2}}}, the given equation is valid, so x = {{{pi/2}}} is a solution. Thus we found that in the first quadrant, 0 <= x <= {{{pi/2}}}, the only solution is x = {{{pi/2}}}, where both sides of the given equation are defined and the equation is valid. In the second quadrant, {{{pi/2}}} < x <= {{{pi}}}, the left side of the given equation is EITHER positive OR not defined, while the right side is always negative, so in the domain {{{pi/2}}} < x <= {{{pi}}} our given equation has no solutions. In QIII and QIV, where {{{pi}}} < x < {{{2pi}}}, (a) sin(x) is negative and the term (b) {{{-sqrt(sin(x)+cos(x))}}} is either negative or not defined. Therefore, the left side of the given equation is not defined at {{{pi}}} < x < {{{2pi}}}. So, in the union { QIII U QIV ) there is no solution to equation (1), at all. Thus we found that the only solution to equation (1) in the interval [{{{0}}},{{{2pi}}}) is x = {{{pi/2}}}. If you want to get the GENERAL solution for any real x, then use the fact that both left side and right side of equation (1) are periodic functions of x with the period {{{2pi}}}. Hence, the general solution to equation (1) is the set of values {{{pi/2 + 2k*pi}}}, where 'k' is any integer k = 0, +/-1, +/-2, . . . and so on. At this point, the problem is solved completely. </pre> The method of solution is an accurate analysis of left side and right side of the equation. My other lessons in this site on plotting and analyzing functions are - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Finding-x-intercepts-and-y-intercepts.lesson>Finding x-intercepts and y-intercepts</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Compressing-and-stretching-of-graphs.lesson>Compressing and stretching graphs</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/HOW-TO-PLOT-transformed-functions.lesson>HOW TO PLOT transformed functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/HOW-TO-write-functions-for-transformed-plots.lesson>HOW TO write functions for transformed plots</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/HOW-TO-PLOT-transformed-periodic-trigonometry-functions.lesson>HOW TO PLOT transformed periodic trigonometry functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Analyzing-periodic-trig-functions-for-amplitude-period-vert-and-hor-shifts.lesson>Analyzing periodic trigonometric functions for the amplitude, the period, vertical and horizontal shifts</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Do-not-fall-into-a-TRAP-when-analysing-problems-on-trigonometric-functions.lesson>Do not fall into a TRAP when analyzing problems on trigonometric functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/The-domain-and-the-range-of-transformed-functions.lesson>The domain and the range of transformed functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Write-a-function-that-has-given-transformations-from-the-parent-function.lesson>Write a function which is a result of given transformations of the parent function</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Describe-transformations-from-the-given-basic-function-to-final-function.lesson>Describe transformations from the given parent function to final function</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Writing-a-function-rule-for-a-function-based-on-its-wording-description.lesson>Writing a function rule for a function based on its wording description</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Constructing-a-function-based-on-its-given-properties.lesson>Constructing a function based on its given properties</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Finding-inverse-functions.lesson>Finding inverse functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Miscellaneous-problems-on-plots-of-functions.lesson>Miscellaneous problems on plots of functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Given-a-point-on-a-plot-of-a-function--find-the-corresponding-point-on-the-plot-of-transformed-function.lesson>Given a point on a plot of a function, find the corresponding point on the plot of transformed function</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Special-advanced-problems-on-finding-the-domain-of-functions.lesson>Special advanced problems on finding the domain of functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Special-advanced-problems-on-finding-the-range-of-functions.lesson>Special advanced problems on finding the range of functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/OVERVIEW-of-lessons-on-plotting-and-analyzing-functions.lesson>OVERVIEW of lessons on plotting and analyzing functions</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I. 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.
