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