Answer
$$7(cos\cdot60+isin\cdot60)\cdot2(cos\cdot30+isin\cdot30)=25200i^4n^2s^2+50400ci^2nos^2+25200c^2o^2s^2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}7(cos\cdot60+isin\cdot60)\cdot2(cos\cdot30+isin\cdot30)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}7(60cos+60i^2ns)\cdot2(30cos+30i^2ns) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(420cos+420i^2ns)\cdot2(30cos+30i^2ns) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(840cos+840i^2ns)(30cos+30i^2ns) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}25200c^2o^2s^2+25200ci^2nos^2+25200ci^2nos^2+25200i^4n^2s^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}25200i^4n^2s^2+50400ci^2nos^2+25200c^2o^2s^2\end{aligned} $$ | |
| ① | $$ i s i n \cdot 60 = 60 i^{1 + 1} n s = 60 i^2 n s $$$$ i s i n \cdot 30 = 30 i^{1 + 1} n s = 30 i^2 n s $$ |
| ② | Multiply $ \color{blue}{7} $ by $ \left( 60cos+60i^2ns\right) $ $$ \color{blue}{7} \cdot \left( 60cos+60i^2ns\right) = 420cos+420i^2ns $$ |
| ③ | $$ \left( \color{blue}{420cos+420i^2ns}\right) \cdot 2 = 840cos+840i^2ns $$ |
| ④ | Multiply each term of $ \left( \color{blue}{840cos+840i^2ns}\right) $ by each term in $ \left( 30cos+30i^2ns\right) $. $$ \left( \color{blue}{840cos+840i^2ns}\right) \cdot \left( 30cos+30i^2ns\right) = \\ = 25200c^2o^2s^2+25200ci^2nos^2+25200ci^2nos^2+25200i^4n^2s^2 $$ |
| ⑤ | Combine like terms: $$ 25200c^2o^2s^2+ \color{blue}{25200ci^2nos^2} + \color{blue}{25200ci^2nos^2} +25200i^4n^2s^2 = 25200i^4n^2s^2+ \color{blue}{50400ci^2nos^2} +25200c^2o^2s^2 $$ |
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