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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
triangles
, or
a random puzzle
.
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prime numbers
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binary
numbers
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dates
regular shapes
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crossnumbers
means
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unit fractions
powers
time
dominos
calculus
multiplication
chess
mean
clocks
wordplay
sport
ellipses
sum to infinity
percentages
coins
the only crossnumber
shapes
consecutive integers
axes
folding tube maps
geometry
squares
expansions
odd numbers
area
tournaments
factorials
triangles
indices
rectangles
factors
dice
2d shapes
division
partitions
geometric means
routes
circles
multiples
square grids
even numbers
polynomials
menace
cards
elections
lists
consecutive numbers
pascal's triangle
star numbers
median
complex numbers
differentiation
cryptic clues
shape
number
fractions
grids
cryptic crossnumbers
triangle numbers
neighbours
coordinates
colouring
algebra
graphs
people maths
geometric mean
scales
hexagons
sums
balancing
probability
money
digital products
albgebra
floors
matrices
advent
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dodecagons
perfect numbers
addition
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quadrilaterals
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tangents
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