maths Flashcards
cosh2a =
2cosh(^2)a -1
1 + 2sinh(^2)a
tanha 2a =
2tanha / 1 + tanh(^2)a
what does 1 equal in hyperbolic functions
cosh(^2)a - sinh(^2)a =1
sech(^2)a=
1 - tanh(^2)a
-cosech(^2)a =
1- coth(^2)a
sinh(x+y) +
sinhxcoshx + sinhycoshx
cosh(x+y) =
coshxcoshy + sinhxsinhy
what does 1 equal in normal trig
cos(^2)a +sin(^2)a = 1
sec(^2)a =
1 +tan(^2)a
cosec(^2)a =
1 +cot(^2)a
sin2a =
2sinacosa
cos2a=
cos(^2)a - sinh(^2)a
2cos(^2)a - 1
1 - 2sin(^2)a
tan2a
= 2tana / 1-tan(^2)a
compex numbers
2cos(nx) =
z(^n) + 1/z(^n)
complex numbers
2isin(nx)
z (^n) - 1/z(^n)
y axis reflection
-1 0
0 1
x axis reflection
1 0
0 -1
y = x reflection
0 1
1 0
y = -x reflection
0 -1
-1 0
2 d rotation
cosx -sinx
sinx cosx
3 x3
x = 0 reflection
-1 0 0
0 1 0
0 0 1
y = 0 reflection 3x3
1 0 0
0 -1 0
0 0 1
reflection z = 0
1 0 0
0 1 0
0 0 -1
rotation about x axis
1 0 0
0 cosx -sinx
0 sinx cosx
roation about y axis
cosx 0 -sinx
0 1 0
sinx 0 cosx
z axis rotation
cosx -sinx 0
sinx cosx 0
0 0 1
What is scalar product
a.b = /a//b/ . cosx
what is scalar product if two lines are parallel
a.b = 0
angle between 2 lines
cos x = /a.b/ divided by /a//b/
angle between line and a plane
sin x = b.n divided by /b//n/
angle between 2 planes
modulus of n.n divided by /n//n/
how to find intersection of plane and line
- write equation of line as one single vector
- subsistutute vector into r.n = d form of plane equation
- solve to find lamda and then substitute lamda into line equation to get coordinates
how to find intersection of two planes along a line
write cartesian from of each equation
eliminate one variable using simultaneous equations and express one variable in terms of the other
substitute the new equation back into original plane equation to get an equation in terms of the original eliminated variable
rewrite the equation found in step 2 and change the subject
then write form of line in cartesian form
perpendicular distance between point and a line
find equation for line
find equation from a to b ( a being the point and b being a general point on the line)
AB is perpendicular to line so AB . direction of line = 0
thatll give you lamda
then sub lamda into AB equation and modulus of it is distance
distance between point and plane
use formula bok
distance between two parralell lines
find BA by doing line 1 - line 2
then set t=lamda - mu
BA is perpendicular to line 1
find t then sub back in and find modulus
