mscroggs
.co.uk
mscroggs
.co.uk
blog
puzzles
academic
talks
contact
subscribe
subscribe
Loading image...
Puzzles
Arctan
Source:
Futility Closet
Prove that \(\arctan(1)+\arctan(2)+\arctan(3)=\pi\).
Show answer & extension
Hide answer & extension
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
2d shapes
, or
a random puzzle
.
Archive
Show me a random puzzle
Most recent collections
Advent calendar 2023
Advent calendar 2022
Advent calendar 2021
Advent calendar 2020
List of all puzzles
Tags
regular shapes
fractions
coins
squares
tangents
graphs
area
albgebra
perimeter
pentagons
triangles
sequences
logic
calculus
addition
star numbers
sport
trigonometry
taxicab geometry
advent
algebra
irreducible numbers
gerrymandering
shapes
division
arrows
ellipses
planes
unit fractions
folding tube maps
chalkdust crossnumber
sums
polynomials
spheres
products
determinants
cubics
balancing
ave
christmas
surds
2d shapes
cryptic crossnumbers
circles
triangle numbers
the only crossnumber
cube numbers
menace
clocks
crosswords
parabolas
multiplication
multiples
geometry
books
median
lines
geometric mean
means
money
proportion
wordplay
routes
expansions
factors
functions
dice
probability
even numbers
rectangles
sets
rugby
square roots
partitions
tournaments
tiling
crossnumber
factorials
axes
range
complex numbers
percentages
combinatorics
chess
remainders
cryptic clues
bases
volume
cards
pascal's triangle
digital clocks
dominos
crossnumbers
grids
chocolate
floors
digital products
numbers
dodecagons
decahedra
integers
quadratics
differentiation
mean
colouring
matrices
sum to infinity
angles
speed
hexagons
probabilty
perfect numbers
binary
doubling
number
games
scales
odd numbers
consecutive numbers
integration
time
coordinates
elections
prime numbers
symmetry
square numbers
averages
dates
digits
consecutive integers
polygons
3d shapes
geometric means
shape
indices
palindromes
quadrilaterals
people maths
Archive
Show me a random puzzle
▼ show ▼
▲ hide ▲
Most recent collections
Advent calendar 2023
Advent calendar 2022
Advent calendar 2021
Advent calendar 2020
List of all puzzles
Tags
surds
prime numbers
floors
square numbers
triangles
games
geometric mean
probabilty
integers
hexagons
books
square roots
crosswords
crossnumber
ellipses
grids
coordinates
averages
cryptic clues
number
geometric means
squares
doubling
consecutive integers
factorials
colouring
graphs
pascal's triangle
tournaments
digital clocks
probability
lines
scales
chocolate
algebra
albgebra
determinants
dominos
chalkdust crossnumber
palindromes
logic
fractions
digital products
decahedra
calculus
multiples
time
numbers
sets
area
speed
menace
2d shapes
spheres
balancing
arrows
planes
mean
proportion
factors
even numbers
quadrilaterals
integration
crossnumbers
division
partitions
median
consecutive numbers
quadratics
shapes
polygons
expansions
triangle numbers
matrices
chess
trigonometry
functions
symmetry
percentages
rugby
routes
sum to infinity
unit fractions
sequences
sums
tangents
addition
differentiation
combinatorics
sport
dice
cryptic crossnumbers
coins
gerrymandering
3d shapes
dates
indices
volume
multiplication
shape
binary
taxicab geometry
star numbers
ave
circles
elections
irreducible numbers
the only crossnumber
geometry
cards
rectangles
clocks
axes
wordplay
cube numbers
folding tube maps
angles
polynomials
people maths
parabolas
pentagons
money
dodecagons
means
products
complex numbers
cubics
tiling
advent
perfect numbers
perimeter
range
odd numbers
digits
christmas
regular shapes
bases
remainders
© Matthew Scroggs 2012–2024