8.5 complex numbers(imaginary)

To divide two complex vectors

- z1/z2=r1/r2(cos(θ1-θ2)+isin(θ1-θ2)) -Abbreviated: z1/z2=r1/r2cis(θ1-θ2)

DeMoivre's Therom

-- therom states that if z=r(cos(theta)+isin(theta)) is a complex number and the n is a positive integer, then z^n=r^n(cosn(theta)+isin n(theta)).

Roots of complex numbers

-Convert complex vector into trig form. -replace z on the one side with r^ncisθn=the complex vector given. -n will be equal to however many roots you need to find. -then take the n root of the r value of the complex vector and that's r. Then divide θ by the number of roots you need to find. -to get the other roots you need to find divide 360 by n and add that to the 1st angle measure you get until you get the amount of roots you need. - sketch a picture -

Equations

-Cos(theta)=a/r -sin(theta)=b/r -a=rcos(theta) -b=rsin(theta) -you can use this knowledge to replace a and b with what it's equal to and get the "complex number in tirg form" -it looks like z=r(cos(theta)+isin(theta) -Remember as z=rcis(theta)

To find θ

-Tanθ=b/a -θ=tan^-1(b/a) -sketch to decide quadrant

Solving for things x when the answer is complex

-Things like x^4+81=0 will be complex. -get 81 to the other side and think of it as a complex vector--> -81+0i. -convert to trig form and find it's roots by equaling it the r^4cisθ4. Then use previous steps explained to solve. -make a sketch.

Represent geometrically

-Y axis is imaginary and x is real. -Form vector into triangle with sides a and b, use Pythagorean therom to find r(the hypotenuse). - the angle is counterclockwise but you can use the angle in the triangle as a reference one to solve.

To solve multiplying and dividing with complex vectors

-put into trig form using previous equations. - sketch a graph - use equations to solve -put into complex form again

To multiply two complex vectors

-z1×z2=r1×r2(cos(θ1+θ2))+isin(θ1+θ2)) -abbreviated:z1×z2=r1×r2cis(θ1+θ2)

Absolute value of a complex number

-|z|=square root of (a^2+b^2) - same as r and magnitude of a vector.

Complex vectors to a power

Convert vector to trig form and use DeMoivre's therom. Expand and solve

Powers of i

It's a pattern. i,-1,-i,1 i=i i^2=-1 i^3=-i i^4=1 And this repeats itself.

Absolute value

The -- of a complex number a+bi is the distance between the origin (0,0) and the point (a,b).

Trig form Modulus Argument

The --- of a complex number z=a+bi is given by z=r(cos(theta)+isin(theta)), where r is the -- of z and theta is the -- of z.

Nth root

The complex number u=a+bi is a -- of the complex number z when z=u^n=(a+bi)^n

DeMoivre's therom

Z^n=rcisθn

Complex form

a+bi a=real Component bi=imaginary Component