FM Pure U4 Flashcards
sin(2a)
2sin(a)cos(a)
cos(2a)
cos^2(a) - sin^2(a)
tan(2a)
2tan(a) / 1-tan^2(a)
General solutions sin=
n(pi) + (-1)^nPV
General solutions cos=
2n(pi) +- PV
General solutions tan=
n(pi) + PV
How to solve asin(x) + bcos(x)
=rsin(x)cos(c) + rcos(x)sin(c)
Where a=rcos(c) and b=rsin(c)
r= sqrt(a^2 + b^2) and c - tan^(-1)(b/a)
Small angles
sin(dx) ~ tan(dx) ~ dx
cos(dx) ~ 1- (1/2)(dx)^2
arg(a + ib)
tan^(-1) (b/a)
De Moivre a+ib =
= r(cos(c) + isin(c))
If z^n = cos(nc) + isin(nc),
z^n + z^-n =
z^n - z^-n =
z^n + z^-n = 2cos(nc)
z^n - z^-n = 2isin(nc)
If z^n = cos(nc) + isin(nc),
2cos(nc) =
2isin(nc) =
2cos(nc) = z^n + z^-n
2isin(nc) = z^n - z^-n
Complex roots of unity often in form. Sum of roots
1, w, w^2. Sum = 0
Z^n = 1 (roots of unity) is given by
formula book
int(1/x)
ln(x) + c
tan^2 + 1 =
sec^2
1 + cot^2 =
cosec^2
sec^2 in terms of tan
tan^2 + 1
cosec^2 in terms of cot
1 + cot^2
Vol of revolution abt x:
pi int(y^2)dx
Vol of revolution abt y
pi int(x^2)dy
cosh(x)
1/2(e^x + e^-x)
sinh(x)
1/2(e^x - e^-x)
Converting trig to hyperbolics, when does sign change?
On product of sines.
How would you deal with int(sqrt(a^2 - x^2))?
Substitute x=asinu
How would you deal with int(a^2 + x^2)?
Substitute x=atanu (or partial fractions, I suppose)
How would you deal with int(sqrt(x^2 - a^2))?
Substitute x=acoshu
How would you deal with int(sqrt(x^2 + a^2))?
Substitute x=asinhu
