Sum of red = Sum of blue?
The figure below is made on the regular grid.
Which is bigger, the red angle, or the sum of the blue angles?
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1 solution
I suppose that last calculation should be tan ( θ b 1 + θ b 2 ) = 1 8 1 7 2 1 = 1 7 9 .
The final result is still the same as 2 1 < 1 7 9 . :)
Assuming each grid side length equals unity, we have:
θ r e d = a r c t a n ( 2 1 ) , θ b l u e 1 = a r c t a n ( 3 1 ) , θ b l u e 2 = a r c t a n ( 6 1 ) .
Knowing that the tangent function is an increasing function, let's now calculate:
t a n ( θ r e d ) = 2 1 .
t a n ( θ b l u e 1 + θ b l u e 2 ) = 1 − t a n ( θ b l u e 1 ) t a n ( θ b l u e 2 ) t a n ( θ b l u e 1 ) + t a n ( θ b l u e 2 ) = 1 − ( 3 1 ) ( 6 1 ) 3 1 + 6 1 = 1 7 9 .
Since 2 1 < 1 7 9 , the blue angle sum is greater.