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