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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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Most recent collections
Advent calendar 2023
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List of all puzzles
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logic
perimeter
ave
geometry
perfect numbers
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cubics
dates
digital products
symmetry
the only crossnumber
addition
chocolate
colouring
factorials
sum to infinity
sets
advent
sequences
complex numbers
palindromes
scales
volume
binary
square roots
3d shapes
digital clocks
area
coins
doubling
lines
tiling
arrows
chess
sport
cryptic crossnumbers
elections
menace
albgebra
planes
triangles
wordplay
probabilty
digits
odd numbers
christmas
matrices
graphs
indices
sums
square numbers
ellipses
integers
pascal's triangle
range
angles
consecutive integers
consecutive numbers
money
cards
even numbers
folding tube maps
spheres
partitions
people maths
dominos
grids
averages
star numbers
tournaments
geometric means
algebra
irreducible numbers
determinants
surds
crossnumber
functions
chalkdust crossnumber
clocks
factors
division
balancing
differentiation
routes
numbers
2d shapes
unit fractions
floors
shapes
crosswords
proportion
hexagons
crossnumbers
geometric mean
time
fractions
quadratics
products
books
multiples
prime numbers
median
taxicab geometry
calculus
circles
polynomials
bases
regular shapes
games
shape
trigonometry
combinatorics
integration
rugby
pentagons
quadrilaterals
parabolas
dodecagons
decahedra
gerrymandering
expansions
rectangles
tangents
cryptic clues
mean
cube numbers
coordinates
number
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speed
means
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