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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
2d shapes
, or
a random puzzle
.
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pascal's triangle
chocolate
colouring
planes
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cubics
regular shapes
range
lines
prime factors
sequences
spheres
coordinates
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albgebra
speed
fractions
addition
menace
money
multiplication
dates
expansions
dodecagons
scales
logic
partitions
digital clocks
circles
differentiation
percentages
squares
the only crossnumber
2d shapes
factors
cube numbers
geometric means
square roots
median
odd numbers
number
shape
decahedra
bases
polygons
tangents
chess
digital products
coins
sum to infinity
tiling
quadratics
mean
probability
integration
multiplaction squares
doubling
consecutive numbers
triangles
gerrymandering
games
angles
lists
polynomials
powers
floors
triangle numbers
determinants
prime numbers
medians
grids
area
even numbers
rugby
sport
indices
functions
division
sets
irreducible numbers
binary
crossnumbers
taxicab geometry
neighbours
time
elections
digits
cryptic clues
combinatorics
christmas
dice
consecutive integers
factorials
square grids
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