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# Having trouble with some Trig and Complex Number Problems

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Overnight, I've been working on some more problems to keep practicing my skills. However, I'm having some trouble with these problems:

Jul 15, 2021

#1
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Here is 7.1.3

This should also help with 7.1.5 if you were struggling to understand the meaning of cis.

For 7.1.4 multiply out the bracketed terms (using foil), then collect real and imaginary terms together, remembering that i^2 = -1.

Jul 15, 2021
#2
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alternate solution to 7.1.3 without knowing euler's formula(as it is only learned in calculus i believe):

$$\frac{w}{z}\\ = \frac{r}{s}\cdot\frac{\cos(\alpha)+i\sin(\alpha)}{\cos{\beta}+i\sin{\beta}}\\ = \frac{r}{s}\cdot\frac{(\cos(\alpha)+i\sin(\alpha))(\cos{\beta}-i\sin{\beta})} {(\cos{\beta}+i\sin{\beta})(\cos{\beta}-i\sin{\beta})}\\ = \frac{r}{s}\cdot\frac{(\cos(\alpha)+i\sin(\alpha))(\cos{\beta}-i\sin{\beta})}{\cos^2{\beta}+\sin^2{\beta}}\\ = \frac{r}{s}\cdot\frac{\cos{a}\cos{b}-i\sin{b}\cos{a}+i\sin{a}\cos{b}+\sin{\beta}\sin{\alpha}}{\cos^2{\beta}+\sin^2{\beta}}$$

by the Pythagorean identity and angle difference rules, this reduces to:

$$\frac{r}{s}\cdot(\cos(\alpha-\beta)-i\sin(\alpha-\beta))\\=\boxed{\frac{r}{s}cis(\alpha-\beta)}$$

Also, for 7.1.4, try converting them into polar form, because it will make it easier to multiply (in your head, of course).

Jul 15, 2021