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Change of coordinates


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1












$begingroup$


enter image description here



Hi everyone!



Can someone explain me this exercice? My solution was $thetain [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$, which is incorrect, but I can't understand why...



The other parts were correct.



I thought $x leq y Leftrightarrow rcdotcos(theta)sin(phi) le rcdot sin(theta)sin(phi) Leftrightarrow 1 le operatornametg(theta)$ so $theta in [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$










share|cite|improve this question











$endgroup$
















    1












    $begingroup$


    enter image description here



    Hi everyone!



    Can someone explain me this exercice? My solution was $thetain [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$, which is incorrect, but I can't understand why...



    The other parts were correct.



    I thought $x leq y Leftrightarrow rcdotcos(theta)sin(phi) le rcdot sin(theta)sin(phi) Leftrightarrow 1 le operatornametg(theta)$ so $theta in [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$










    share|cite|improve this question











    $endgroup$














      1












      1








      1





      $begingroup$


      enter image description here



      Hi everyone!



      Can someone explain me this exercice? My solution was $thetain [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$, which is incorrect, but I can't understand why...



      The other parts were correct.



      I thought $x leq y Leftrightarrow rcdotcos(theta)sin(phi) le rcdot sin(theta)sin(phi) Leftrightarrow 1 le operatornametg(theta)$ so $theta in [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$










      share|cite|improve this question











      $endgroup$




      enter image description here



      Hi everyone!



      Can someone explain me this exercice? My solution was $thetain [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$, which is incorrect, but I can't understand why...



      The other parts were correct.



      I thought $x leq y Leftrightarrow rcdotcos(theta)sin(phi) le rcdot sin(theta)sin(phi) Leftrightarrow 1 le operatornametg(theta)$ so $theta in [fracpi4,fracpi2]cup[frac5pi4,frac3pi2)$







      calculus






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 2 days ago









      Ingix

      4,867159




      4,867159










      asked 2 days ago









      aluno20000aluno20000

      446




      446




















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          You forgot that dividing (and multiplying) an inequality by some number requires checking the sign of the divisor (or factor), in order to find out if the inequality changes the direction.



          That's because from $a le b$ it surely does not follow $-a le -b$, the opppsite is true: $a le b Leftrightarrow -a ge -b$.



          That doesn't affect the common factor $sin(phi)$, as that is non-negative for $0 le phi < pi$. But $cos(theta)$ changes signs at $theta=fracpi2,frac3pi2$. That means when you divide by it to get the inequality involving $tan(theta)$ you need to consider 2 cases:



          1) $cos(theta) ge 0$, or $theta in [0,pi over 2] cup [3pi over 2,2pi)$. Your correct solution set for $tan(theta) ge 1$ needs then be restricted to the given interval of $theta$, which gives $I_1(theta)=[pi over 4,pi over 2]$.



          2) $cos(theta) < 0$, or $theta in (pi over 2, 3pi over 2)$. In that interval, then the inequality becomes $tan(theta) le 1$. The solution set of this, restricted to $(pi over 2, 3pi over 2)$ is $I_2(theta)=(pi over 2,5pi over 4]$.



          Finally, the possible set for $theta$ is $I_1(theta) cup I_2(theta) = [pi over 4,5pi over 4]$, as given in the book.



          So the main thing to remember is that multiplying/dividing an inequality requires extra steps to get the correct direction of the inequality!






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Thank you for the answer!
            $endgroup$
            – aluno20000
            2 days ago










          Your Answer





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          1 Answer
          1






          active

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          1 Answer
          1






          active

          oldest

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          active

          oldest

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          active

          oldest

          votes









          1












          $begingroup$

          You forgot that dividing (and multiplying) an inequality by some number requires checking the sign of the divisor (or factor), in order to find out if the inequality changes the direction.



          That's because from $a le b$ it surely does not follow $-a le -b$, the opppsite is true: $a le b Leftrightarrow -a ge -b$.



          That doesn't affect the common factor $sin(phi)$, as that is non-negative for $0 le phi < pi$. But $cos(theta)$ changes signs at $theta=fracpi2,frac3pi2$. That means when you divide by it to get the inequality involving $tan(theta)$ you need to consider 2 cases:



          1) $cos(theta) ge 0$, or $theta in [0,pi over 2] cup [3pi over 2,2pi)$. Your correct solution set for $tan(theta) ge 1$ needs then be restricted to the given interval of $theta$, which gives $I_1(theta)=[pi over 4,pi over 2]$.



          2) $cos(theta) < 0$, or $theta in (pi over 2, 3pi over 2)$. In that interval, then the inequality becomes $tan(theta) le 1$. The solution set of this, restricted to $(pi over 2, 3pi over 2)$ is $I_2(theta)=(pi over 2,5pi over 4]$.



          Finally, the possible set for $theta$ is $I_1(theta) cup I_2(theta) = [pi over 4,5pi over 4]$, as given in the book.



          So the main thing to remember is that multiplying/dividing an inequality requires extra steps to get the correct direction of the inequality!






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Thank you for the answer!
            $endgroup$
            – aluno20000
            2 days ago















          1












          $begingroup$

          You forgot that dividing (and multiplying) an inequality by some number requires checking the sign of the divisor (or factor), in order to find out if the inequality changes the direction.



          That's because from $a le b$ it surely does not follow $-a le -b$, the opppsite is true: $a le b Leftrightarrow -a ge -b$.



          That doesn't affect the common factor $sin(phi)$, as that is non-negative for $0 le phi < pi$. But $cos(theta)$ changes signs at $theta=fracpi2,frac3pi2$. That means when you divide by it to get the inequality involving $tan(theta)$ you need to consider 2 cases:



          1) $cos(theta) ge 0$, or $theta in [0,pi over 2] cup [3pi over 2,2pi)$. Your correct solution set for $tan(theta) ge 1$ needs then be restricted to the given interval of $theta$, which gives $I_1(theta)=[pi over 4,pi over 2]$.



          2) $cos(theta) < 0$, or $theta in (pi over 2, 3pi over 2)$. In that interval, then the inequality becomes $tan(theta) le 1$. The solution set of this, restricted to $(pi over 2, 3pi over 2)$ is $I_2(theta)=(pi over 2,5pi over 4]$.



          Finally, the possible set for $theta$ is $I_1(theta) cup I_2(theta) = [pi over 4,5pi over 4]$, as given in the book.



          So the main thing to remember is that multiplying/dividing an inequality requires extra steps to get the correct direction of the inequality!






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Thank you for the answer!
            $endgroup$
            – aluno20000
            2 days ago













          1












          1








          1





          $begingroup$

          You forgot that dividing (and multiplying) an inequality by some number requires checking the sign of the divisor (or factor), in order to find out if the inequality changes the direction.



          That's because from $a le b$ it surely does not follow $-a le -b$, the opppsite is true: $a le b Leftrightarrow -a ge -b$.



          That doesn't affect the common factor $sin(phi)$, as that is non-negative for $0 le phi < pi$. But $cos(theta)$ changes signs at $theta=fracpi2,frac3pi2$. That means when you divide by it to get the inequality involving $tan(theta)$ you need to consider 2 cases:



          1) $cos(theta) ge 0$, or $theta in [0,pi over 2] cup [3pi over 2,2pi)$. Your correct solution set for $tan(theta) ge 1$ needs then be restricted to the given interval of $theta$, which gives $I_1(theta)=[pi over 4,pi over 2]$.



          2) $cos(theta) < 0$, or $theta in (pi over 2, 3pi over 2)$. In that interval, then the inequality becomes $tan(theta) le 1$. The solution set of this, restricted to $(pi over 2, 3pi over 2)$ is $I_2(theta)=(pi over 2,5pi over 4]$.



          Finally, the possible set for $theta$ is $I_1(theta) cup I_2(theta) = [pi over 4,5pi over 4]$, as given in the book.



          So the main thing to remember is that multiplying/dividing an inequality requires extra steps to get the correct direction of the inequality!






          share|cite|improve this answer









          $endgroup$



          You forgot that dividing (and multiplying) an inequality by some number requires checking the sign of the divisor (or factor), in order to find out if the inequality changes the direction.



          That's because from $a le b$ it surely does not follow $-a le -b$, the opppsite is true: $a le b Leftrightarrow -a ge -b$.



          That doesn't affect the common factor $sin(phi)$, as that is non-negative for $0 le phi < pi$. But $cos(theta)$ changes signs at $theta=fracpi2,frac3pi2$. That means when you divide by it to get the inequality involving $tan(theta)$ you need to consider 2 cases:



          1) $cos(theta) ge 0$, or $theta in [0,pi over 2] cup [3pi over 2,2pi)$. Your correct solution set for $tan(theta) ge 1$ needs then be restricted to the given interval of $theta$, which gives $I_1(theta)=[pi over 4,pi over 2]$.



          2) $cos(theta) < 0$, or $theta in (pi over 2, 3pi over 2)$. In that interval, then the inequality becomes $tan(theta) le 1$. The solution set of this, restricted to $(pi over 2, 3pi over 2)$ is $I_2(theta)=(pi over 2,5pi over 4]$.



          Finally, the possible set for $theta$ is $I_1(theta) cup I_2(theta) = [pi over 4,5pi over 4]$, as given in the book.



          So the main thing to remember is that multiplying/dividing an inequality requires extra steps to get the correct direction of the inequality!







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 2 days ago









          IngixIngix

          4,867159




          4,867159











          • $begingroup$
            Thank you for the answer!
            $endgroup$
            – aluno20000
            2 days ago
















          • $begingroup$
            Thank you for the answer!
            $endgroup$
            – aluno20000
            2 days ago















          $begingroup$
          Thank you for the answer!
          $endgroup$
          – aluno20000
          2 days ago




          $begingroup$
          Thank you for the answer!
          $endgroup$
          – aluno20000
          2 days ago

















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