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