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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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2d shapes
, or
a random puzzle
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graphs
even numbers
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colouring
bases
3d shapes
probability
angles
rectangles
odd numbers
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spheres
determinants
cryptic clues
perfect numbers
dice
pascal's triangle
quadratics
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numbers
irreducible numbers
lists
shapes
taxicab geometry
sport
parabolas
wordplay
multiplication
geometry
area
circles
doubling
squares
quadrilaterals
speed
volume
coins
cards
integers
ellipses
matrices
polynomials
balancing
cube numbers
powers
tournaments
triangle numbers
square numbers
sets
probabilty
multiplaction squares
sequences
hexagons
crossnumbers
dates
range
sums
combinatorics
regular shapes
digital clocks
gerrymandering
planes
indices
games
arrows
time
cubics
folding tube maps
2d shapes
geometric mean
dodecagons
neighbours
axes
advent
means
menace
xor
polygons
scales
crosswords
palindromes
shape
rugby
factorials
partitions
books
proportion
tiling
integration
people maths
lines
functions
median
factors
coordinates
averages
chalkdust crossnumber
ave
pentagons
unit fractions
chess
products
prime factors
calculus
square grids
binary
routes
division
trigonometry
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christmas
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consecutive integers
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