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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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Archive
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▼ show ▼
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Most recent collections
Advent calendar 2023
Advent calendar 2022
Advent calendar 2021
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List of all puzzles
Tags
decahedra
wordplay
crosswords
balancing
area
dominos
scales
squares
indices
rugby
symmetry
grids
tiling
polynomials
numbers
cryptic crossnumbers
triangles
lines
consecutive numbers
prime numbers
sport
trigonometry
square numbers
dodecagons
quadrilaterals
advent
shape
palindromes
chess
cards
cryptic clues
differentiation
chocolate
expansions
parabolas
digital products
ave
sums
coins
money
determinants
irreducible numbers
pentagons
clocks
the only crossnumber
volume
dates
algebra
complex numbers
integration
even numbers
factors
crossnumbers
tournaments
perimeter
remainders
bases
quadratics
menace
geometry
multiplication
median
tangents
geometric mean
spheres
factorials
probability
angles
percentages
colouring
calculus
star numbers
speed
rectangles
products
coordinates
folding tube maps
time
circles
taxicab geometry
sequences
consecutive integers
dice
cube numbers
geometric means
people maths
integers
books
partitions
ellipses
range
planes
number
binary
doubling
elections
3d shapes
cubics
mean
crossnumber
addition
polygons
hexagons
surds
2d shapes
probabilty
perfect numbers
square roots
gerrymandering
axes
sets
proportion
digital clocks
floors
means
graphs
fractions
combinatorics
christmas
logic
regular shapes
pascal's triangle
triangle numbers
chalkdust crossnumber
arrows
division
odd numbers
digits
averages
functions
shapes
games
unit fractions
matrices
albgebra
routes
multiples
sum to infinity
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