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Puzzles
Arctan
Source:
Futility Closet
Prove that \(\arctan(1)+\arctan(2)+\arctan(3)=\pi\).
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Let \(\alpha=\arctan(1)\), \(\beta=\arctan(2)\) and \(\gamma=\arctan(3)\), then draw the angles as follows:
Then proceed as in
Three Squares
.
Extension
Can you find any other integers \(a\), \(b\) and \(c\) such that:
$$\arctan(a)+\arctan(b)+\arctan(c)=\pi$$
Tags:
geometry
,
2d shapes
,
triangles
,
trigonometry
If you enjoyed this puzzle, check out
Sunday Afternoon Maths XXXVIII
,
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sum to infinity
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algebra
colouring
volume
rugby
complex numbers
shape
digital clocks
numbers
chocolate
2d shapes
quadrilaterals
logic
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dice
unit fractions
gerrymandering
calculus
mean
division
sequences
probabilty
prime numbers
prime factors
sums
surds
balancing
digital products
chalkdust crossnumber
addition
symmetry
odd numbers
geometry
triangles
pascal's triangle
arrows
planes
axes
people maths
indices
rectangles
routes
cryptic clues
tangents
palindromes
integration
functions
dominos
determinants
consecutive numbers
quadratics
triangle numbers
books
irreducible numbers
range
area
fractions
coins
probability
3d shapes
coordinates
lines
partitions
speed
factors
powers
polygons
games
albgebra
pentagons
neighbours
advent
digits
tournaments
xor
parabolas
cards
spheres
elections
expansions
dodecagons
crossnumbers
remainders
geometric means
scales
consecutive integers
taxicab geometry
doubling
shapes
time
integers
square grids
squares
the only crossnumber
decahedra
products
tiling
floors
proportion
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