Algebra Question

Question

Simplify, leaving answer in polar form.

cos(5A) - i sin(5A) / i sin(B) + cos(B)

(division)

Answer 1

Note the i on the first term.

This is simply e^-5Ai/e^Bi = e^(-5A-B)i = cos(-5A-B) + i sin(-5A-B) = cos(5A+B) - i sin(5A+B), as cos is even and sin is odd.

You could also solve this problem by turning the division into a multiplication; you will realize that the complex conjugate of cos(B)+ i sin(B), cos(B) - i sin(B), is actually its reciprocal, as cos² + sin² = 1, and when you start multiplying, you will recognize that the real term is the cos addition formula and the imaginary term is the sin addition formula.