I have a question related to complex number representation of sinusoids from a book by Steven W. Smith "Digital Signal Processing". The book is made freely available by the author on dspguide.com. The part I have question about is available here, pp.559-563.
Excerpt from the book (p. 560):
A sinusoid is represented by a complex number in rectangular form as follows:
$$ A \cos(\omega t) + B \sin(\omega t) \rightleftharpoons a + jb $$
where $a \rightleftharpoons A$ and $b \rightleftharpoons -B$.
A sinusoid is represented by a complex number in polar form as follows:
$$ M \cos(\omega t + \phi) \rightleftharpoons M e^{j\theta} $$
where $\theta \rightleftharpoons -\phi$.
Why change the sign of the imaginary part and phase angle? This is to make the substitution appear in the same form as the complex Fourier transform described in the next chapter. The substitution techniques of this chapter gain nothing from this sign change, but it is almost always done to keep things consistent with the more advanced methods.
What I am confused about is that these two representations are not equivalent. Here is an example following the rules described above for rectangular and polar forms:
$$ \cos(\omega t) + \sin(\omega t) \rightleftharpoons 1 - j = \sqrt{2} e^{-j\frac{\pi}{4}} $$
$$ \cos(\omega t) + \sin(\omega t) = \sqrt{2} \cos(\omega t - \frac{\pi}{4}) \rightleftharpoons \sqrt{2} e^{j\frac{\pi}{4}} = 1 + j $$
The rules for rectangular and polar form produce phase angles of opposite sign.
Author gives few more examples on p. 561 (first paragraph in "Complex Represenation of Systems") that show clear discrepancy between rectangular and polar forms considering phase angle.
I am aware that we can take either rectangular or polar form to "be correct", the sign is a convention anyway. But how can both of them be used at the same time?
