B9 Exponential & cis fns - Vector and Complex Numbers:

YOU are better than YOU think. Show yourself how:

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In mathematics, sooner or later you need to learn to read like a lawyer. For that read logic chapters 1 to 5 in Three Skills for Algebra. Sooner is better. Good luck.

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On the phone with a classmate or tutor,
or twiddla to write & draw with each other on art, math & science etc.

Logic Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying, Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Do not leave here without it - Logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

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Caution: Site advice is approximately correct, for some circumstances, not all. Site How-TOs are logically developed, but not tried and tested. That leaves room for thought and refinement..

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After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;

For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

The cis or exponential functions

cis(q) = cos(q)+isin(q) = exp(iq)

cis(A)·cis(B) = cis(A+B) or equivalenly
exp(iA)·exp(iB) = exp(i(A+B))

add the angles, multiply the lengths

The notation exp(iA) is employed because it is possible to define the exponential exp(z) of a complex number z = a + ib where a and b are both real. The definition for those of you who know about exponentials exp(a) of real numbers a is as follows:

exp(a + ib) = exp(x) {cos(b) + i sin(b)).

For more details visit Calculus and Beyond.

Many trig identities follow from the above property (exponential property) and the equality of two different ways to compute products of complex numbers.

Cosine and Sine Addition Formulas

Preliminary Exercise: In the identity exp(iA)·exp(iB) = exp(i(A+B)), express the left hand side in terms of the real and imaginary parts of the factors.

Solution:

exp(iA)·exp(iB) = {cos(A) + i sin (A)} {cos(B) + i sin (B))

= cos(A)cos(B)-sin(A)sin(B) + i{cos(A)sin(B) + sin(A)cos(B)}

exp(i(A+B)) = cos(A+B) + i sin(A+B).

The solution to the above exercise, implies the cosine and sine addition formulas:

cos(A+B) = cos(A)cos(B)-sin(A)sin(B)

sin (A+B) = cos(A)sin(B) + sin(A)cos(B)