Question

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

Answer 1

`0.6306+2.716i`

Step-by-step explanation

`\begin{gathered} \lbrack(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(1)/(4)(\cos 10+isen10)\rbrack\text{ }\lbrack\text{1}2(\cos 75+i\sin 75) \\ \lbrack(1)/(4)\cos 10+(1)/(4)isen10\rbrack\lbrack\text{ 1}2\cos 75+12i\sin 75\rbrack \\ ((1)/(4)\cos 10\cdot\text{ 1}2\cos 75)+((1)/(4)\cos 10\cdot12i\sin 75)+((1)/(4)isen10\cdot\text{1}2\cos 75)+((1)/(4)isen10\cdot12i\sin 75) \\ (12)/(4)\cos 10\cos 75+(12)/(4)i\cos 10\sin 75+(12)/(4)i\text{sen}10\cos 75+(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