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
trigonometry
, 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
mean
balancing
cubics
quadratics
consecutive numbers
rectangles
multiples
square numbers
bases
odd numbers
quadrilaterals
rugby
time
calculus
complex numbers
gerrymandering
trigonometry
range
axes
chess
advent
floors
irreducible numbers
median
tiling
prime numbers
palindromes
polygons
people maths
integration
polynomials
crossnumber
digital products
probability
factorials
speed
symmetry
routes
binary
sum to infinity
pentagons
logic
doubling
sequences
angles
geometric mean
expansions
multiplication
addition
remainders
shape
ave
geometry
coins
books
digits
differentiation
proportion
sets
partitions
planes
means
even numbers
spheres
crossnumbers
pascal's triangle
ellipses
cube numbers
area
integers
determinants
3d shapes
menace
averages
arrows
lines
fractions
matrices
geometric means
volume
indices
shapes
colouring
functions
division
consecutive integers
algebra
square roots
2d shapes
coordinates
christmas
factors
dominos
scales
cards
hexagons
folding tube maps
triangles
wordplay
numbers
triangle numbers
circles
elections
crosswords
taxicab geometry
perfect numbers
star numbers
graphs
number
dice
products
clocks
cryptic clues
money
squares
cryptic crossnumbers
unit fractions
dodecagons
parabolas
percentages
combinatorics
tournaments
probabilty
albgebra
dates
decahedra
games
surds
sport
digital clocks
chalkdust crossnumber
regular shapes
grids
sums
perimeter
tangents
chocolate
the only crossnumber
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
multiples
indices
coins
calculus
combinatorics
sum to infinity
dates
trigonometry
the only crossnumber
averages
matrices
fractions
sums
addition
squares
rectangles
numbers
star numbers
people maths
bases
digits
3d shapes
ave
products
geometric means
scales
cryptic crossnumbers
expansions
arrows
multiplication
games
consecutive numbers
area
graphs
remainders
quadratics
median
spheres
irreducible numbers
quadrilaterals
dominos
taxicab geometry
crossnumber
palindromes
triangles
square roots
mean
cards
2d shapes
algebra
prime numbers
triangle numbers
polynomials
probabilty
surds
shape
routes
chess
polygons
digital clocks
cube numbers
parabolas
advent
range
tournaments
partitions
albgebra
rugby
grids
perimeter
ellipses
dodecagons
tiling
chocolate
means
regular shapes
angles
binary
wordplay
pentagons
lines
tangents
planes
factors
unit fractions
clocks
volume
cryptic clues
pascal's triangle
colouring
decahedra
percentages
digital products
symmetry
functions
folding tube maps
integers
differentiation
factorials
doubling
speed
dice
floors
complex numbers
circles
geometric mean
crosswords
chalkdust crossnumber
gerrymandering
hexagons
money
books
number
balancing
sequences
elections
cubics
even numbers
odd numbers
sets
coordinates
probability
shapes
square numbers
logic
integration
menace
determinants
consecutive integers
sport
crossnumbers
proportion
perfect numbers
geometry
division
time
christmas
axes
© Matthew Scroggs 2012–2024