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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
,
puzzles about
trigonometry
, or
a random puzzle
.
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graphs
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folding tube maps
perfect numbers
addition
percentages
remainders
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gerrymandering
christmas
spheres
matrices
square roots
dominos
volume
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cards
even numbers
complex numbers
ellipses
factors
fractions
unit fractions
chalkdust crossnumber
geometric means
division
tangents
xor
medians
angles
irreducible numbers
factorials
triangles
powers
colouring
arrows
geometry
scales
sums
doubling
quadrilaterals
trigonometry
tournaments
decahedra
median
pascal's triangle
money
star numbers
planes
time
coins
binary
probabilty
polynomials
albgebra
sets
triangle numbers
sport
means
neighbours
products
dice
indices
tiling
lines
balancing
lists
crossnumbers
chess
prime factors
crosswords
shapes
symmetry
routes
combinatorics
menace
rugby
advent
proportion
multiples
geometric mean
number
numbers
integers
sequences
shape
wordplay
mean
chocolate
consecutive integers
dodecagons
grids
calculus
integration
partitions
sum to infinity
cubics
polygons
pentagons
floors
books
cube numbers
square grids
surds
rectangles
cryptic crossnumbers
hexagons
the only crossnumber
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squares
differentiation
perimeter
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taxicab geometry
regular shapes
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