1.
_____
z^n = |z|^ncis(n0)

2.
_____
z1 = a+bi
z2 = c+di

z1+z2 = (a+c)+(b+d)i
z1-z2 = (a-c)+(b-d)i
z1*z2 = ac + adi + bci - bd
z1/z2 = (a+bi)*(c-di)/
        (c+di)*(c-di)
_____

3.
_____
z1 = |z1|cis0
z2 = |z2|cis0

z1*z2 = (|z1|z2|)cis(0+0)
z1/z2 = (|z1/z2|)cis(0-0)

4.
_____
sin^20+cos^20 = 1
tan^20+1      = sec^20
cot^20+1      = cosec^20

5.
_____
tan-1(√3)) =π/3
tan-1(1/√3)=π/6
tan-1(1)   =π/4

6.
____
sin(a+-b)=sin(a)cos(b)+- cos(a)sin(b)
cos(a+-b)=cos(a)cos(b)-+ sin(a)sin(b)
tan(a+-b)=tan(a)+-tan(b)/
       1 -+ tan(a)tan(b)

7.
____
sec(x)=1/cos(x)
csc(x)=1/sin(x)
cot(x)=1/tan(x)=Cos/Sin
tan(x)=1/cot(x)=Sin/Cos

8.
____
(cos0+sin0)^n -> use fibonachi
for proofs
n = 2, 121
    3, 1331
    4, 14641

9.
____
z^n=x
z^n=xcisπ
z^n=xcisπ+2kπ
z  =(xcisπ+2kπ)^1/n
-->
z = xcis1/n(π+fkπ/m)
z = xcisπ/nm(1+fk)

10.
____
Express asinx+bcosx in form
Asin(x+0)
Acos0sinx+Asin0cosx = asinx+-bcosx
rearange
Acos0 = a, Asin0 = b.

A^2 = A^2 + 1
A^2 = A^2(cos^20+sin^20)
    = √a^2+b^2

tan0 = Asin0/Acos0
   0 = tan-1(b/a)

asinx+bcosx =
A sin (x + tan-1(b/a))
A cos (x + tan-1(-a/b))

asinx-bcosx =
A sin (x + tan-1(-b/a))
A cos (x + tan-1(a/b))

Sketch asin(x+c)
a = amp, ps = -c.

1.
____
Transformation
(a c)(x) where x,y is
(b d)(y) vector
where a,b is x Line
where c,d is y Line

2.
____
Finding Multiple
(T3)(T2)(T1)(V)
(T3)->(T2)->(T1)
(T123) is Tm

3.
____
Identity
(1 0)
(0 1)
ReflectionX(Mx)
(1  0)
(0 -1)
ReflectionY(My)
(-1 0)
( 0 1)
ReflectionY=x(My=x)
(0 1)
(1 0)
Dilation(Enlargment)=x(Dk)
(k 0)
(0 k)
Rotation(R0)
(cos0 -sin0)
(sin0  cos0)

4.
____
Translations
Xi + Yj

5.
____
When transformation is (a,b)
y=x
sub vector
-> (y+b) = (x+a)
-> let transformation be (a,b)
if solving..

6.
____
When transformations are like
Ti = ai+bj
Tj = ci+dj
Tm = (a c)
     (b d)

7.
____
When transfomation is Matrix
(x') (a b)(x)
(y')=(c d)(y)
where x,y is vec,
x'y' is image.

x'= ax+by and
y'= cx+dy

(x')(a b)-1 (x)
(y')(c d)  =(y)