Why is this angle not $22.5^circ$? And does it have an exact value?












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$begingroup$


Since the angle which splits a square in a half, starting from it's bottom left corner, is $45^circ$, I intuitively thought that, if I put two squares to be horizontally adjacent, the angle between the bottom side of the resulting rectangle and the line going from it's bottom left vertex to it's top right vertex would be $22.5^circ$. My (flawed) reasoning was that it does "half the vertical space a $45^circ$ angle does". It looks like it's not though, as it's somewhere around $27^circ$.



enter image description here



I'm sorry if it's a lame question, but why is that? And does that angle have an exact value that we can mathematically derive?






trigonometry angle







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$endgroup$

















    1












    $begingroup$


    Since the angle which splits a square in a half, starting from it's bottom left corner, is $45^circ$, I intuitively thought that, if I put two squares to be horizontally adjacent, the angle between the bottom side of the resulting rectangle and the line going from it's bottom left vertex to it's top right vertex would be $22.5^circ$. My (flawed) reasoning was that it does "half the vertical space a $45^circ$ angle does". It looks like it's not though, as it's somewhere around $27^circ$.



    enter image description here



    I'm sorry if it's a lame question, but why is that? And does that angle have an exact value that we can mathematically derive?






    trigonometry angle







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      Since the angle which splits a square in a half, starting from it's bottom left corner, is $45^circ$, I intuitively thought that, if I put two squares to be horizontally adjacent, the angle between the bottom side of the resulting rectangle and the line going from it's bottom left vertex to it's top right vertex would be $22.5^circ$. My (flawed) reasoning was that it does "half the vertical space a $45^circ$ angle does". It looks like it's not though, as it's somewhere around $27^circ$.



      enter image description here



      I'm sorry if it's a lame question, but why is that? And does that angle have an exact value that we can mathematically derive?






      trigonometry angle







      share|cite|improve this question











      $endgroup$




      Since the angle which splits a square in a half, starting from it's bottom left corner, is $45^circ$, I intuitively thought that, if I put two squares to be horizontally adjacent, the angle between the bottom side of the resulting rectangle and the line going from it's bottom left vertex to it's top right vertex would be $22.5^circ$. My (flawed) reasoning was that it does "half the vertical space a $45^circ$ angle does". It looks like it's not though, as it's somewhere around $27^circ$.



      enter image description here



      I'm sorry if it's a lame question, but why is that? And does that angle have an exact value that we can mathematically derive?






      trigonometry angle




      trigonometry angle






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 6 at 9:53









      Eevee Trainer

      7,61621338




      7,61621338










      asked Jan 6 at 9:35









      Eärendil BagginsEärendil Baggins

      1156




      1156






















          2 Answers
          2






          active

          oldest

          votes


















          5












          $begingroup$

          The core reason that it is not $22.5^circ$, essentially, is that the trigonometric functions are not linear functions. For example, aside from particular values of $x$,



          $$sin(2x) neq 2 cdot sin(x)$$
          $$sin(x+y) neq sin(x) + sin(y)$$



          (Similar truths hold for the other functions.) Thus, you cannot expect to get half the angle just by doubling the length of a side of a triangle.



          As for the angle in question, it can shown to be given by $tan^{-1}(1/2)$:



          enter image description here



          For all my looking, this expression doesn't seem to have an exact value, and is probably irrational. A variety of representations (e.g. continued fractions, integrals, infinite sums) can be found through Wolfram Alpha.



          As for approximations,



          $$tan^{-1}(1/2) approx 26.57^circ approx 0.4636476 text{ radians}$$






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Great answer! Only one detail I'd like to understand better: what kind of reasoning do we employ to know that the value is arctan(1/2)? Does it have to do with the fact that the angle leads the line defining it to intersect the opposite side of the square in the 1/2 position? Sorry for my awful mathematical terminology.
            $endgroup$
            – Eärendil Baggins
            Jan 6 at 9:53






          • 2




            $begingroup$
            It's basically because that diagonal - the line you make through the two squares - essentially divides the rectangle into two right triangles. Opposite the angle you're concerned with, the side length is $1$, and the adjacent side is $2$, so per definition of the tangent function, when $x$ is the angle in question, we say $tan(x) = 1/2$. Then, taking the inverse tangent of both sides, $x = tan^{-1}(1/2)$.
            $endgroup$
            – Eevee Trainer
            Jan 6 at 9:56






          • 3




            $begingroup$
            You have a right-angled triangle with legs of 1 and 2 units. By definition of the tangent function, the pointiest angle has tangent $1/2$. @EärendilBaggins
            $endgroup$
            – Lord Shark the Unknown
            Jan 6 at 9:57



















          5












          $begingroup$

          The angle is $tan^{-1}(1/2)$. The tangent of $22.5$ degrees is $sqrt2-1$
          which is about $0.414$ so the actual angle is a bit more than $22.5$ degrees.






          share|cite|improve this answer









          $endgroup$













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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            5












            $begingroup$

            The core reason that it is not $22.5^circ$, essentially, is that the trigonometric functions are not linear functions. For example, aside from particular values of $x$,



            $$sin(2x) neq 2 cdot sin(x)$$
            $$sin(x+y) neq sin(x) + sin(y)$$



            (Similar truths hold for the other functions.) Thus, you cannot expect to get half the angle just by doubling the length of a side of a triangle.



            As for the angle in question, it can shown to be given by $tan^{-1}(1/2)$:



            enter image description here



            For all my looking, this expression doesn't seem to have an exact value, and is probably irrational. A variety of representations (e.g. continued fractions, integrals, infinite sums) can be found through Wolfram Alpha.



            As for approximations,



            $$tan^{-1}(1/2) approx 26.57^circ approx 0.4636476 text{ radians}$$






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              Great answer! Only one detail I'd like to understand better: what kind of reasoning do we employ to know that the value is arctan(1/2)? Does it have to do with the fact that the angle leads the line defining it to intersect the opposite side of the square in the 1/2 position? Sorry for my awful mathematical terminology.
              $endgroup$
              – Eärendil Baggins
              Jan 6 at 9:53






            • 2




              $begingroup$
              It's basically because that diagonal - the line you make through the two squares - essentially divides the rectangle into two right triangles. Opposite the angle you're concerned with, the side length is $1$, and the adjacent side is $2$, so per definition of the tangent function, when $x$ is the angle in question, we say $tan(x) = 1/2$. Then, taking the inverse tangent of both sides, $x = tan^{-1}(1/2)$.
              $endgroup$
              – Eevee Trainer
              Jan 6 at 9:56






            • 3




              $begingroup$
              You have a right-angled triangle with legs of 1 and 2 units. By definition of the tangent function, the pointiest angle has tangent $1/2$. @EärendilBaggins
              $endgroup$
              – Lord Shark the Unknown
              Jan 6 at 9:57
















            5












            $begingroup$

            The core reason that it is not $22.5^circ$, essentially, is that the trigonometric functions are not linear functions. For example, aside from particular values of $x$,



            $$sin(2x) neq 2 cdot sin(x)$$
            $$sin(x+y) neq sin(x) + sin(y)$$



            (Similar truths hold for the other functions.) Thus, you cannot expect to get half the angle just by doubling the length of a side of a triangle.



            As for the angle in question, it can shown to be given by $tan^{-1}(1/2)$:



            enter image description here



            For all my looking, this expression doesn't seem to have an exact value, and is probably irrational. A variety of representations (e.g. continued fractions, integrals, infinite sums) can be found through Wolfram Alpha.



            As for approximations,



            $$tan^{-1}(1/2) approx 26.57^circ approx 0.4636476 text{ radians}$$






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              Great answer! Only one detail I'd like to understand better: what kind of reasoning do we employ to know that the value is arctan(1/2)? Does it have to do with the fact that the angle leads the line defining it to intersect the opposite side of the square in the 1/2 position? Sorry for my awful mathematical terminology.
              $endgroup$
              – Eärendil Baggins
              Jan 6 at 9:53






            • 2




              $begingroup$
              It's basically because that diagonal - the line you make through the two squares - essentially divides the rectangle into two right triangles. Opposite the angle you're concerned with, the side length is $1$, and the adjacent side is $2$, so per definition of the tangent function, when $x$ is the angle in question, we say $tan(x) = 1/2$. Then, taking the inverse tangent of both sides, $x = tan^{-1}(1/2)$.
              $endgroup$
              – Eevee Trainer
              Jan 6 at 9:56






            • 3




              $begingroup$
              You have a right-angled triangle with legs of 1 and 2 units. By definition of the tangent function, the pointiest angle has tangent $1/2$. @EärendilBaggins
              $endgroup$
              – Lord Shark the Unknown
              Jan 6 at 9:57














            5












            5








            5





            $begingroup$

            The core reason that it is not $22.5^circ$, essentially, is that the trigonometric functions are not linear functions. For example, aside from particular values of $x$,



            $$sin(2x) neq 2 cdot sin(x)$$
            $$sin(x+y) neq sin(x) + sin(y)$$



            (Similar truths hold for the other functions.) Thus, you cannot expect to get half the angle just by doubling the length of a side of a triangle.



            As for the angle in question, it can shown to be given by $tan^{-1}(1/2)$:



            enter image description here



            For all my looking, this expression doesn't seem to have an exact value, and is probably irrational. A variety of representations (e.g. continued fractions, integrals, infinite sums) can be found through Wolfram Alpha.



            As for approximations,



            $$tan^{-1}(1/2) approx 26.57^circ approx 0.4636476 text{ radians}$$






            share|cite|improve this answer











            $endgroup$



            The core reason that it is not $22.5^circ$, essentially, is that the trigonometric functions are not linear functions. For example, aside from particular values of $x$,



            $$sin(2x) neq 2 cdot sin(x)$$
            $$sin(x+y) neq sin(x) + sin(y)$$



            (Similar truths hold for the other functions.) Thus, you cannot expect to get half the angle just by doubling the length of a side of a triangle.



            As for the angle in question, it can shown to be given by $tan^{-1}(1/2)$:



            enter image description here



            For all my looking, this expression doesn't seem to have an exact value, and is probably irrational. A variety of representations (e.g. continued fractions, integrals, infinite sums) can be found through Wolfram Alpha.



            As for approximations,



            $$tan^{-1}(1/2) approx 26.57^circ approx 0.4636476 text{ radians}$$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Jan 6 at 9:58

























            answered Jan 6 at 9:50









            Eevee TrainerEevee Trainer

            7,61621338




            7,61621338












            • $begingroup$
              Great answer! Only one detail I'd like to understand better: what kind of reasoning do we employ to know that the value is arctan(1/2)? Does it have to do with the fact that the angle leads the line defining it to intersect the opposite side of the square in the 1/2 position? Sorry for my awful mathematical terminology.
              $endgroup$
              – Eärendil Baggins
              Jan 6 at 9:53






            • 2




              $begingroup$
              It's basically because that diagonal - the line you make through the two squares - essentially divides the rectangle into two right triangles. Opposite the angle you're concerned with, the side length is $1$, and the adjacent side is $2$, so per definition of the tangent function, when $x$ is the angle in question, we say $tan(x) = 1/2$. Then, taking the inverse tangent of both sides, $x = tan^{-1}(1/2)$.
              $endgroup$
              – Eevee Trainer
              Jan 6 at 9:56






            • 3




              $begingroup$
              You have a right-angled triangle with legs of 1 and 2 units. By definition of the tangent function, the pointiest angle has tangent $1/2$. @EärendilBaggins
              $endgroup$
              – Lord Shark the Unknown
              Jan 6 at 9:57


















            • $begingroup$
              Great answer! Only one detail I'd like to understand better: what kind of reasoning do we employ to know that the value is arctan(1/2)? Does it have to do with the fact that the angle leads the line defining it to intersect the opposite side of the square in the 1/2 position? Sorry for my awful mathematical terminology.
              $endgroup$
              – Eärendil Baggins
              Jan 6 at 9:53






            • 2




              $begingroup$
              It's basically because that diagonal - the line you make through the two squares - essentially divides the rectangle into two right triangles. Opposite the angle you're concerned with, the side length is $1$, and the adjacent side is $2$, so per definition of the tangent function, when $x$ is the angle in question, we say $tan(x) = 1/2$. Then, taking the inverse tangent of both sides, $x = tan^{-1}(1/2)$.
              $endgroup$
              – Eevee Trainer
              Jan 6 at 9:56






            • 3




              $begingroup$
              You have a right-angled triangle with legs of 1 and 2 units. By definition of the tangent function, the pointiest angle has tangent $1/2$. @EärendilBaggins
              $endgroup$
              – Lord Shark the Unknown
              Jan 6 at 9:57
















            $begingroup$
            Great answer! Only one detail I'd like to understand better: what kind of reasoning do we employ to know that the value is arctan(1/2)? Does it have to do with the fact that the angle leads the line defining it to intersect the opposite side of the square in the 1/2 position? Sorry for my awful mathematical terminology.
            $endgroup$
            – Eärendil Baggins
            Jan 6 at 9:53




            $begingroup$
            Great answer! Only one detail I'd like to understand better: what kind of reasoning do we employ to know that the value is arctan(1/2)? Does it have to do with the fact that the angle leads the line defining it to intersect the opposite side of the square in the 1/2 position? Sorry for my awful mathematical terminology.
            $endgroup$
            – Eärendil Baggins
            Jan 6 at 9:53




            2




            2




            $begingroup$
            It's basically because that diagonal - the line you make through the two squares - essentially divides the rectangle into two right triangles. Opposite the angle you're concerned with, the side length is $1$, and the adjacent side is $2$, so per definition of the tangent function, when $x$ is the angle in question, we say $tan(x) = 1/2$. Then, taking the inverse tangent of both sides, $x = tan^{-1}(1/2)$.
            $endgroup$
            – Eevee Trainer
            Jan 6 at 9:56




            $begingroup$
            It's basically because that diagonal - the line you make through the two squares - essentially divides the rectangle into two right triangles. Opposite the angle you're concerned with, the side length is $1$, and the adjacent side is $2$, so per definition of the tangent function, when $x$ is the angle in question, we say $tan(x) = 1/2$. Then, taking the inverse tangent of both sides, $x = tan^{-1}(1/2)$.
            $endgroup$
            – Eevee Trainer
            Jan 6 at 9:56




            3




            3




            $begingroup$
            You have a right-angled triangle with legs of 1 and 2 units. By definition of the tangent function, the pointiest angle has tangent $1/2$. @EärendilBaggins
            $endgroup$
            – Lord Shark the Unknown
            Jan 6 at 9:57




            $begingroup$
            You have a right-angled triangle with legs of 1 and 2 units. By definition of the tangent function, the pointiest angle has tangent $1/2$. @EärendilBaggins
            $endgroup$
            – Lord Shark the Unknown
            Jan 6 at 9:57











            5












            $begingroup$

            The angle is $tan^{-1}(1/2)$. The tangent of $22.5$ degrees is $sqrt2-1$
            which is about $0.414$ so the actual angle is a bit more than $22.5$ degrees.






            share|cite|improve this answer









            $endgroup$


















              5












              $begingroup$

              The angle is $tan^{-1}(1/2)$. The tangent of $22.5$ degrees is $sqrt2-1$
              which is about $0.414$ so the actual angle is a bit more than $22.5$ degrees.






              share|cite|improve this answer









              $endgroup$
















                5












                5








                5





                $begingroup$

                The angle is $tan^{-1}(1/2)$. The tangent of $22.5$ degrees is $sqrt2-1$
                which is about $0.414$ so the actual angle is a bit more than $22.5$ degrees.






                share|cite|improve this answer









                $endgroup$



                The angle is $tan^{-1}(1/2)$. The tangent of $22.5$ degrees is $sqrt2-1$
                which is about $0.414$ so the actual angle is a bit more than $22.5$ degrees.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 6 at 9:40









                Lord Shark the UnknownLord Shark the Unknown

                105k1161133




                105k1161133






























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