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