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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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rugby
remainders
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prime factors
surds
medians
games
taxicab geometry
gerrymandering
irreducible numbers
shape
proportion
advent
square roots
chess
perfect numbers
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sums
dice
ave
christmas
indices
multiples
digits
percentages
fractions
dates
routes
cards
decahedra
wordplay
unit fractions
albgebra
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speed
multiplaction squares
binary
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2d shapes
factorials
sum to infinity
range
sequences
spheres
factors
digital clocks
quadrilaterals
dominos
money
trigonometry
powers
books
grids
colouring
addition
neighbours
triangles
probability
integers
the only crossnumber
geometric mean
square grids
doubling
3d shapes
odd numbers
crossnumbers
chalkdust crossnumber
balancing
integration
tiling
number
hexagons
bases
logic
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calculus
combinatorics
palindromes
differentiation
angles
complex numbers
geometric means
determinants
means
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numbers
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