$p_0^2 + q_0^2 = 1$ and $-sin(s) = -p_0sin(s) + q_0cos(s)$ $rightarrow $ $p_0 = -cos2s, q_0 = -sin(2s)$












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How does solving $p_0^2 + q_0^2 = 1$ and $-sin(s) = -p_0sin(s) + q_0cos(s)$ give $p_0 = -cos2s $ and $q_0 = -sin(2s)$?



I can clearly see one solution is $p_0=1,q_0=0$ but can't seem to understand the other.










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$endgroup$

















    0












    $begingroup$


    How does solving $p_0^2 + q_0^2 = 1$ and $-sin(s) = -p_0sin(s) + q_0cos(s)$ give $p_0 = -cos2s $ and $q_0 = -sin(2s)$?



    I can clearly see one solution is $p_0=1,q_0=0$ but can't seem to understand the other.










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      How does solving $p_0^2 + q_0^2 = 1$ and $-sin(s) = -p_0sin(s) + q_0cos(s)$ give $p_0 = -cos2s $ and $q_0 = -sin(2s)$?



      I can clearly see one solution is $p_0=1,q_0=0$ but can't seem to understand the other.










      share|cite|improve this question











      $endgroup$




      How does solving $p_0^2 + q_0^2 = 1$ and $-sin(s) = -p_0sin(s) + q_0cos(s)$ give $p_0 = -cos2s $ and $q_0 = -sin(2s)$?



      I can clearly see one solution is $p_0=1,q_0=0$ but can't seem to understand the other.







      trigonometry systems-of-equations






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 17 '18 at 12:24









      Harry Peter

      5,46911439




      5,46911439










      asked Dec 10 '18 at 14:41









      pablo_mathscobarpablo_mathscobar

      996




      996






















          2 Answers
          2






          active

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          0












          $begingroup$

          HINT : Let $p_0=cos x $ and $q_0=sin x$. Then the second equation becomes $$-sin s = -cos xsin s + sin xcos s.$$
          That is
          $$-sin s = sin(x-s).$$
          Now solve for $x$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I still cant seem to do it
            $endgroup$
            – pablo_mathscobar
            Dec 10 '18 at 14:56










          • $begingroup$
            Another hint:$sin A+sin B=0 iff 2sinfrac{A+B}2cosfrac{A-B}2=0$. This means either $sinfrac{A+B}2=0$ or $cosfrac{A-B}2=0$.
            $endgroup$
            – Thomas Shelby
            Dec 10 '18 at 15:12



















          0












          $begingroup$

          Since $p_0^2 + q_0^2 = 1$, there is $theta$ such that $p_0 = cos(theta)$ and $q_0 = sin(theta)$. Thus your equation becomes
          $$ sin(-s) = -sin(s) = - cos(theta) sin(s) + sin(theta) cos(s) = sin(theta - s)$$
          Thus either $theta = 2 n pi$ or $theta = (2n+1)pi + 2 s$.






          share|cite|improve this answer









          $endgroup$













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            2 Answers
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            active

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            2 Answers
            2






            active

            oldest

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            active

            oldest

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            active

            oldest

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            0












            $begingroup$

            HINT : Let $p_0=cos x $ and $q_0=sin x$. Then the second equation becomes $$-sin s = -cos xsin s + sin xcos s.$$
            That is
            $$-sin s = sin(x-s).$$
            Now solve for $x$.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              I still cant seem to do it
              $endgroup$
              – pablo_mathscobar
              Dec 10 '18 at 14:56










            • $begingroup$
              Another hint:$sin A+sin B=0 iff 2sinfrac{A+B}2cosfrac{A-B}2=0$. This means either $sinfrac{A+B}2=0$ or $cosfrac{A-B}2=0$.
              $endgroup$
              – Thomas Shelby
              Dec 10 '18 at 15:12
















            0












            $begingroup$

            HINT : Let $p_0=cos x $ and $q_0=sin x$. Then the second equation becomes $$-sin s = -cos xsin s + sin xcos s.$$
            That is
            $$-sin s = sin(x-s).$$
            Now solve for $x$.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              I still cant seem to do it
              $endgroup$
              – pablo_mathscobar
              Dec 10 '18 at 14:56










            • $begingroup$
              Another hint:$sin A+sin B=0 iff 2sinfrac{A+B}2cosfrac{A-B}2=0$. This means either $sinfrac{A+B}2=0$ or $cosfrac{A-B}2=0$.
              $endgroup$
              – Thomas Shelby
              Dec 10 '18 at 15:12














            0












            0








            0





            $begingroup$

            HINT : Let $p_0=cos x $ and $q_0=sin x$. Then the second equation becomes $$-sin s = -cos xsin s + sin xcos s.$$
            That is
            $$-sin s = sin(x-s).$$
            Now solve for $x$.






            share|cite|improve this answer









            $endgroup$



            HINT : Let $p_0=cos x $ and $q_0=sin x$. Then the second equation becomes $$-sin s = -cos xsin s + sin xcos s.$$
            That is
            $$-sin s = sin(x-s).$$
            Now solve for $x$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 10 '18 at 14:50









            Thomas ShelbyThomas Shelby

            2,877421




            2,877421












            • $begingroup$
              I still cant seem to do it
              $endgroup$
              – pablo_mathscobar
              Dec 10 '18 at 14:56










            • $begingroup$
              Another hint:$sin A+sin B=0 iff 2sinfrac{A+B}2cosfrac{A-B}2=0$. This means either $sinfrac{A+B}2=0$ or $cosfrac{A-B}2=0$.
              $endgroup$
              – Thomas Shelby
              Dec 10 '18 at 15:12


















            • $begingroup$
              I still cant seem to do it
              $endgroup$
              – pablo_mathscobar
              Dec 10 '18 at 14:56










            • $begingroup$
              Another hint:$sin A+sin B=0 iff 2sinfrac{A+B}2cosfrac{A-B}2=0$. This means either $sinfrac{A+B}2=0$ or $cosfrac{A-B}2=0$.
              $endgroup$
              – Thomas Shelby
              Dec 10 '18 at 15:12
















            $begingroup$
            I still cant seem to do it
            $endgroup$
            – pablo_mathscobar
            Dec 10 '18 at 14:56




            $begingroup$
            I still cant seem to do it
            $endgroup$
            – pablo_mathscobar
            Dec 10 '18 at 14:56












            $begingroup$
            Another hint:$sin A+sin B=0 iff 2sinfrac{A+B}2cosfrac{A-B}2=0$. This means either $sinfrac{A+B}2=0$ or $cosfrac{A-B}2=0$.
            $endgroup$
            – Thomas Shelby
            Dec 10 '18 at 15:12




            $begingroup$
            Another hint:$sin A+sin B=0 iff 2sinfrac{A+B}2cosfrac{A-B}2=0$. This means either $sinfrac{A+B}2=0$ or $cosfrac{A-B}2=0$.
            $endgroup$
            – Thomas Shelby
            Dec 10 '18 at 15:12











            0












            $begingroup$

            Since $p_0^2 + q_0^2 = 1$, there is $theta$ such that $p_0 = cos(theta)$ and $q_0 = sin(theta)$. Thus your equation becomes
            $$ sin(-s) = -sin(s) = - cos(theta) sin(s) + sin(theta) cos(s) = sin(theta - s)$$
            Thus either $theta = 2 n pi$ or $theta = (2n+1)pi + 2 s$.






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              Since $p_0^2 + q_0^2 = 1$, there is $theta$ such that $p_0 = cos(theta)$ and $q_0 = sin(theta)$. Thus your equation becomes
              $$ sin(-s) = -sin(s) = - cos(theta) sin(s) + sin(theta) cos(s) = sin(theta - s)$$
              Thus either $theta = 2 n pi$ or $theta = (2n+1)pi + 2 s$.






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                Since $p_0^2 + q_0^2 = 1$, there is $theta$ such that $p_0 = cos(theta)$ and $q_0 = sin(theta)$. Thus your equation becomes
                $$ sin(-s) = -sin(s) = - cos(theta) sin(s) + sin(theta) cos(s) = sin(theta - s)$$
                Thus either $theta = 2 n pi$ or $theta = (2n+1)pi + 2 s$.






                share|cite|improve this answer









                $endgroup$



                Since $p_0^2 + q_0^2 = 1$, there is $theta$ such that $p_0 = cos(theta)$ and $q_0 = sin(theta)$. Thus your equation becomes
                $$ sin(-s) = -sin(s) = - cos(theta) sin(s) + sin(theta) cos(s) = sin(theta - s)$$
                Thus either $theta = 2 n pi$ or $theta = (2n+1)pi + 2 s$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 10 '18 at 14:51









                Robert IsraelRobert Israel

                322k23212465




                322k23212465






























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