Question

Multiply the two complex numbers. Leave your answer in trigonometric form. Show all work.

Answer 1

$0.6306+2.716i$

Explanation

$\begin{gathered} \lbrack\frac{1}{4}(\cos 10+isen10)\rbrack\text{ }2(\cos 75+i\sin 75) \\ \end{gathered}$

Step 1

apply distributive property.

remember

$(a+b)(c+d)=ac+ad+bc+bd$

so

$\begin{gathered} \lbrack\frac{1}{4}(\cos 10+isen10)\rbrack\text{ }\lbrack\text{1}2(\cos 75+i\sin 75) \\ \lbrack\frac{1}{4}\cos 10+\frac{1}{4}isen10\rbrack\lbrack\text{ 1}2\cos 75+12i\sin 75\rbrack \\ (\frac{1}{4}\cos 10\cdot\text{ 1}2\cos 75)+(\frac{1}{4}\cos 10\cdot12i\sin 75)+(\frac{1}{4}isen10\cdot\text{1}2\cos 75)+(\frac{1}{4}isen10\cdot12i\sin 75) \\ \frac{12}{4}\cos 10\cos 75+\frac{12}{4}i\cos 10\sin 75+\frac{12}{4}i\text{sen}10\cos 75+\frac{12}{4}\text{sen}10\cos 75(i^2) \\ 3\cos 10\cos 75+3i\cos 10\sin 75+3i\sin 10\cos 75+3\sin 10\cos (75)\cdot(-1) \\ 3\cos 10\cos 75+3i\cos 10\sin 75+3i\sin 10\cos 75-3\sin 10\cos (75) \\ \text{add like terms} \\ 3\cos 10\cos 75-3\sin 10\cos (75)+3i\cos 10\sin 75+3i\sin 10\cos 75 \\ (3\cos 10\cos 75-3\sin 10\cos (75))+i(3\cos 10\sin 75+3\sin 10\cos 75) \\ \\ \end{gathered}$

therefore, the answer is

$\begin{gathered} (3\cos 10\cos 75-3\sin 10\cos (75))+(3\cos 10\sin 75+3\sin 10\cos 75)i \\ (0.7646-0.134)+(2.85+0.134)i \\ 0.6306+2.716i \end{gathered}$$0.6306+2.716i$

I hope this helps you