Further Pure

Different forms of Z^n=

r^n(cos(nx)+isin(nx))

(rcisx)^n= r^n cis(nx)

[r, x]^n=[r^n,nx]

Where nx= arg, r^n=mod

Smallest pos integer n for which z^n is imaginary

arg(nx)= pi/2 , 3pi/2

Smallest pos integer n for which z^n is real

arg(nx)= pi, 2pi

Eulers formula

re^ix = r(cos x + i sinx)

Cartesian form of eulers formula

x+iy

4^(3+2i) in eulers form

4³ (e^ln(4)^(2i))

= 4³(cos(ln16)+isin(ln16))

z+z*=

2rcosx

z-z*=

i2rsinx

z x z*=

r²

cosx=

cos(x+2pik)

sin x=

sin(x+2pik)

cosx +i sinx=

e^i(x+2pik)

Sn=

a(r^(n) -1)/(r-1)

Nth roots of unity are

1, e^(i2pi/n), e^(i4pi/n),…,e^(i2(n-1)pi/n)

W_k=W^k_1=

(e^(i2pi/n))^k

w_(n-1)=

w_(n-2)=

…

=W_1*

=W_2*

…

Multiplying by

r(cos x+ i sin x)

e^irx

Means to translate:

Rotate by x, enlarge by r

Dividing by

r(cos x+ i sin x)

e^irx

Means to translate:

Rotate -x, enlarge by 1/r

Cos nx=

(c+is)^n

e^inx+e^-inx=

2cos (nx)

e^inx-e-inx=

2isin(nx)

Z^n+1/z^n

2cos(nx)

z^n-1/z^n=

2isin(nx)

Conditions for a converging geometric sequence are

-1<r<1

S infinity=

a/(1-r)

Polar coordinate

(r, x)

r is distance from origin, x is angle with initial line

r=k As a polar coordinate

Is a circle

x=k as a polar coordinate is

a half-line

cartesian → polar

X=

Y=

X²+Y²=

rcosx

rsinx

r²

r= sin/cos(nx)

What is n

n is the number of loops

dont draw r<0

looped has

2 tangents

a<b

cardioid has

1 tangent

a=b

dimpled

r>0

b<a<2b

convex/egg

r>0

a=>2b

Regular sin graph

0, pi, 2pi

Regular cos graph

pi/2, 3/2pi

Calculus for polar curves

rmax=

dr/dtheta=0

Area enclosed by polar curve=

½/ r² dtheta

Normally measure between tangents, unless r>0 for any value, then between 2pi and 0

Parallel to initial line

dy/dtheta=0

Perp to the initial line

dx/dtheta=0

/secx tanx

[secx]

/sec²x

[tanx]

tan²x=

sec²x-1

cosec²x=

cot²x+1

Proving De Moivres theorem

Proof by induction for r(cosx+isinx)^n

Evaluate x+iy/a+bi

-convert to mod arg form

-subtract args

-subtract mod powers

-make theta between -pi and pi

X³+bx²+cx+d=0

Find b c and d

b is sum of the roots = -b/a

c is product pairs = c/a

d is product= -d/a

Find all roots of z^n= x+iy

Z^n= re^i(x+2pi k), kEZ

k=0,1,2,-1,-2… for -pi<x<pi

Show that cos_ + cos_ + cos_ = ½

Z^n= x+iy

Z^n-x-iy=0

Roots of z^n=0

Real: cos_ +cos_ +cos_= ½

Find the roots of 1+z+z²+z³=0

a=1

r=z

n= 4

z^n-1=0

z^n=e^i(x+2 pi k) k=0,1,-1,2,-2…

Roots of z^n=p

z=p^1/n

360/n= theta

So z= p^1/n+ p^1/n e^itheta…

z^n=p , find the exact value of cos_

Find roots of z^n=p

Rewrite as z^n-p=0

Analyse for sum/ product/ pairs of the equation to equal 0

i.e z³+1=0 has no z² so sum of roots=0

Find c

State 2 conjugate relationships between the roots of z^n=1

z= 1, e^ix, e^i2x, e^i3x, e^i4x

Look for angles that are equal when one is subtracted from 2pi

“Let w_1 be the conjugate of w_4”

w_1=w_4*

w_2=w_3*

Show that 1/(e^xi -1)= -1/2-i/2 cot (x/2)

(x/2) in final answer is the half angle of x

- 1/(e^xi -1) multiply top and bottom by e^-x/2i, a change in sign for the power as it is a half angle in the final answer

Simplify to get the final answer

Write z^n+p as a product of 2 real quadratic factors

z^n+p=0

Find roots, z= z1, z2, z3

z^n+p= (z-z1)(z-z2)(z-z3)

Multiply out to get 2 quad factors

z^n+p= (z²-az+b)(z²+az+b)

Describe the transformation that maps a onto b (complex numbers)

a=x+iy b=a+bi

a

Mod: 1

arg: pi

b

Mod:2

arg: 2pi

Mod 1→ mod 2 is enlargement 2 and arg pi→ arg 2 pi is rotation pi

so az=b

a(2e^ipi)=b

Find the roots of the regular polygon

-inner

z=p

z^number of points=n

(z^no of points - p)=0

-repeat for outer shape

Multiply the two results to find an equation whose roots will give the roots for the two shapes on the diagram

Use de moivres to derive a formula

Find cos(nx)

(Cos x + i sinx)^n

(c+is)^n

Pascals expansion

R//

Use de moivres to derive a formula

Find cos³x

Use either e^ix +e^-ix= 2cos x

e^ix-e^-ix= 2isinx

In this case (2cosx)³= (e^ix +e^-ix)³

Expand both sides, keeping right side in terms of e

Factorise terms of e to put back into trigonometric form

Integrate cos^6x

Use de moivres

Evaluate

Infinity

E cos(kx)

k=0

Starts writing terms by subbing in k=0,1,2,3…

Find the pattern and write in eulers form

Multiply denominator and rearrange to get answer

Show that (e^ix +1)^n= acos²(x/2)sin(x/2)

Expand to see list of results

Im [(e^ix +1)^n]

Since double angle in answer, take out a factor of e^(ix/2). A double angle only counts if it is the final angle on the list of expanded results, divided by 2.

Expand the e^ix

Write in modulus arg form to get answer

Draw the curve r= 5-2sin(3x)

Draw the sin graph

Find the values for 1 section

Mirror it

Check w graphical calc

Draw curve r= 2-2cos(2x)

Solve for r=0

Draw graph

Plot points

Check w calc, be wary of where r<0

Draw curve 3/2-sinx

sinx max = 1

sinx min = -1

So rmax 3/2-1= 3 when sinx= 1, x= pi/2

So r min 3/2+1= 1 when sinx=-1 x= 3pi/2

Plot points

Draw curve r= x(x-pi)(x-2pi) pretend this x is theta

Draw the cartesian graph with y axis equal to r and x axis equal to angle

y= x(x-pi)(x-2pi)

View the points and their locations

Plot

Find the max and min r for the polar curve

(Can sketch)

Write the curve in cartesian form

Find the max and min value of y in cartesian form

Sketch the polar graph to confirm their location being max/min

Find the max and min r for the polar curve

(Unable to sketch)

rmax= dr/dtheta=0

Show line of symmetry

(r, theta) → (r, -theta)