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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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Most recent collections
Advent calendar 2024
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List of all puzzles
Tags
time
determinants
albgebra
unit fractions
surds
powers
number
division
sets
axes
geometric mean
factorials
combinatorics
odd numbers
pascal's triangle
square numbers
spheres
numbers
addition
factors
neighbours
triangle numbers
sum to infinity
median
bases
even numbers
menace
calculus
polygons
cryptic clues
graphs
cards
christmas
square roots
circles
area
gerrymandering
colouring
decahedra
quadratics
angles
3d shapes
range
chess
quadrilaterals
games
geometric means
rugby
crosswords
sums
prime numbers
palindromes
multiples
partitions
means
rectangles
floors
hexagons
remainders
taxicab geometry
algebra
symmetry
digital products
probability
cubics
elections
cube numbers
routes
advent
square grids
speed
shape
folding tube maps
ellipses
tangents
planes
numbers grids
functions
probabilty
consecutive numbers
medians
tournaments
expansions
logic
chocolate
multiplication
regular shapes
integration
differentiation
lines
parabolas
grids
dodecagons
dates
squares
triangles
crossnumbers
the only crossnumber
people maths
trigonometry
arrows
sequences
dice
star numbers
averages
volume
geometry
sport
chalkdust crossnumber
perimeter
coordinates
proportion
percentages
irreducible numbers
scales
books
binary
polynomials
shapes
consecutive integers
ave
balancing
indices
products
integers
tiling
2d shapes
coins
dominos
clocks
complex numbers
pentagons
cryptic crossnumbers
matrices
money
fractions
mean
perfect numbers
wordplay
digital clocks
doubling
digits
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