Trigonometric inequalities with substitution











up vote
0
down vote

favorite












Using inequality $tan frac{x}{2} > frac{x}{2}$ prove that $sin x > x- frac{x^3}{4}$



I tried with substitution $tan frac{x}{2} = t$



$sin x = frac{2t}{t^2+1}$



$t>frac{x}{2}$



$2t>x$



$t^2+1>1+frac{x^2}{4}$



After this I am unable to do it
Please help me
Thank you






trigonometry







share|cite|improve this question
























  • NB both the original and derived identities are only valid for $x > 0$.
    – Travis
    Nov 13 at 9:55















up vote
0
down vote

favorite












Using inequality $tan frac{x}{2} > frac{x}{2}$ prove that $sin x > x- frac{x^3}{4}$



I tried with substitution $tan frac{x}{2} = t$



$sin x = frac{2t}{t^2+1}$



$t>frac{x}{2}$



$2t>x$



$t^2+1>1+frac{x^2}{4}$



After this I am unable to do it
Please help me
Thank you






trigonometry







share|cite|improve this question
























  • NB both the original and derived identities are only valid for $x > 0$.
    – Travis
    Nov 13 at 9:55













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Using inequality $tan frac{x}{2} > frac{x}{2}$ prove that $sin x > x- frac{x^3}{4}$



I tried with substitution $tan frac{x}{2} = t$



$sin x = frac{2t}{t^2+1}$



$t>frac{x}{2}$



$2t>x$



$t^2+1>1+frac{x^2}{4}$



After this I am unable to do it
Please help me
Thank you






trigonometry







share|cite|improve this question















Using inequality $tan frac{x}{2} > frac{x}{2}$ prove that $sin x > x- frac{x^3}{4}$



I tried with substitution $tan frac{x}{2} = t$



$sin x = frac{2t}{t^2+1}$



$t>frac{x}{2}$



$2t>x$



$t^2+1>1+frac{x^2}{4}$



After this I am unable to do it
Please help me
Thank you






trigonometry




trigonometry






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 13 at 9:39









MathFun123

459216




459216










asked Nov 13 at 9:34









Ashwini

11




11












  • NB both the original and derived identities are only valid for $x > 0$.
    – Travis
    Nov 13 at 9:55


















  • NB both the original and derived identities are only valid for $x > 0$.
    – Travis
    Nov 13 at 9:55
















NB both the original and derived identities are only valid for $x > 0$.
– Travis
Nov 13 at 9:55




NB both the original and derived identities are only valid for $x > 0$.
– Travis
Nov 13 at 9:55










1 Answer
1






active

oldest

votes

















up vote
0
down vote













We know that, $$1-t^4<1$$



$Longrightarrow frac{1}{1+t^2}>1-t^2$



$Longrightarrow frac{2t}{1+t^2}>2t-2t^3>2(frac{x}{2})-2(frac{x}{2})^3$



$Longrightarrow sin(x)>x-frac{x^3}{4}$



Hence proved.



Hope it helps:)






share|cite|improve this answer





















    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














     

    draft saved


    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996526%2ftrigonometric-inequalities-with-substitution%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    0
    down vote













    We know that, $$1-t^4<1$$



    $Longrightarrow frac{1}{1+t^2}>1-t^2$



    $Longrightarrow frac{2t}{1+t^2}>2t-2t^3>2(frac{x}{2})-2(frac{x}{2})^3$



    $Longrightarrow sin(x)>x-frac{x^3}{4}$



    Hence proved.



    Hope it helps:)






    share|cite|improve this answer

























      up vote
      0
      down vote













      We know that, $$1-t^4<1$$



      $Longrightarrow frac{1}{1+t^2}>1-t^2$



      $Longrightarrow frac{2t}{1+t^2}>2t-2t^3>2(frac{x}{2})-2(frac{x}{2})^3$



      $Longrightarrow sin(x)>x-frac{x^3}{4}$



      Hence proved.



      Hope it helps:)






      share|cite|improve this answer























        up vote
        0
        down vote










        up vote
        0
        down vote









        We know that, $$1-t^4<1$$



        $Longrightarrow frac{1}{1+t^2}>1-t^2$



        $Longrightarrow frac{2t}{1+t^2}>2t-2t^3>2(frac{x}{2})-2(frac{x}{2})^3$



        $Longrightarrow sin(x)>x-frac{x^3}{4}$



        Hence proved.



        Hope it helps:)






        share|cite|improve this answer












        We know that, $$1-t^4<1$$



        $Longrightarrow frac{1}{1+t^2}>1-t^2$



        $Longrightarrow frac{2t}{1+t^2}>2t-2t^3>2(frac{x}{2})-2(frac{x}{2})^3$



        $Longrightarrow sin(x)>x-frac{x^3}{4}$



        Hence proved.



        Hope it helps:)







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 13 at 9:45









        Crazy for maths

        4948




        4948






























             

            draft saved


            draft discarded



















































             


            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996526%2ftrigonometric-inequalities-with-substitution%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Probability when a professor distributes a quiz and homework assignment to a class of n students.

            Image
            0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
            Read more

            Aardman Animations

            Image
            Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
            Read more

            Are they similar matrix

            Image
            0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
            Read more