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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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trigonometry
, or
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.
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Advent calendar 2024
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List of all puzzles
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medians
crossnumbers
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products
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probabilty
multiplication
digital products
prime numbers
crossnumber
geometric means
means
cryptic crossnumbers
doubling
averages
wordplay
polynomials
number
sport
regular shapes
speed
functions
fractions
planes
median
sets
balancing
clocks
dodecagons
digits
algebra
geometry
square grids
pascal's triangle
perimeter
polygons
tangents
lines
books
chess
differentiation
quadratics
square numbers
2d shapes
complex numbers
dates
cryptic clues
sequences
tournaments
multiples
sum to infinity
ellipses
powers
christmas
percentages
star numbers
irreducible numbers
crosswords
3d shapes
mean
taxicab geometry
perfect numbers
square roots
trigonometry
colouring
determinants
probability
ave
parabolas
geometric mean
bases
time
spheres
shape
area
consecutive integers
chocolate
integration
surds
factorials
even numbers
routes
elections
rugby
circles
dice
chalkdust crossnumber
combinatorics
indices
triangles
sums
range
menace
volume
scales
angles
binary
matrices
tiling
partitions
decahedra
unit fractions
calculus
money
palindromes
coins
logic
cube numbers
proportion
cubics
people maths
integers
coordinates
folding tube maps
remainders
hexagons
the only crossnumber
factors
floors
expansions
grids
games
rectangles
advent
division
quadrilaterals
odd numbers
arrows
symmetry
addition
dominos
graphs
neighbours
digital clocks
gerrymandering
numbers
consecutive numbers
pentagons
cards
axes
triangle numbers
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