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