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Three unit squares are joined side by side to form a 3 by 1 rectangle, and two diagonals are joined as shown above. Find the acute angle formed by the diagonals.

 

 May 13, 2020
 #1
avatar+111360 
+1

The angle  formed  by  the top  left of the figure  and shorter red line can be found as 

 

arctan (1/2)

 

Similarly, the angle formed by the  top of the  figure and  the longer red line  can be found as

 

arctan (1/3)

 

Using an obscure  trig identity  

 

arctan (u)  +  arctan (v)   =  arctan  [ ( u + v)  / (1 - uv) ]

 

arctan (1/2) + arctan (1/3)  =  arctan  [   (1/2 + 1/3)  / ( 1 - (1/2)(1/3) ) ]  =

 

arctan [ (5/6) / (5/6)  ]  =

 

artan (1)  =   45°  =  the angle  we need

 

 

cool cool cool

 May 13, 2020
edited by CPhill  May 13, 2020
 #2
avatar+23552 
+1

arctan 1/2   + arctan 1/3   = 45 degrees

 May 13, 2020

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