Answer
$$(2cos\cdot20+2isin\cdot20)^3=64000i^6n^3s^3+192000ci^4n^2os^3+192000c^2i^2no^2s^3+64000c^3o^3s^3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(2cos\cdot20+2isin\cdot20)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(40cos+40i^2ns)^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}64000c^3o^3s^3+192000c^2i^2no^2s^3+192000ci^4n^2os^3+64000i^6n^3s^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}64000i^6n^3s^3+192000ci^4n^2os^3+192000c^2i^2no^2s^3+64000c^3o^3s^3\end{aligned} $$ | |
| ① | $$ 2 c o s \cdot 20 = 40 c o s $$ |
| ② | $$ 2 i s i n \cdot 20 = 40 i^{1 + 1} n s = 40 i^2 n s $$ |
| ③ | Find $ \left(40cos+40i^2ns\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 40cos $ and $ B = 40i^2ns $. $$ \left(40cos+40i^2ns\right)^3 = \left( 40cos \right)^3+3 \cdot \left( 40cos \right)^2 \cdot 40i^2ns + 3 \cdot 40cos \cdot \left( 40i^2ns \right)^2+\left( 40i^2ns \right)^3 = 64000c^3o^3s^3+192000c^2i^2no^2s^3+192000ci^4n^2os^3+64000i^6n^3s^3 $$ |
| ④ | Combine like terms: $$ 64000i^6n^3s^3+192000ci^4n^2os^3+192000c^2i^2no^2s^3+64000c^3o^3s^3 = 64000i^6n^3s^3+192000ci^4n^2os^3+192000c^2i^2no^2s^3+64000c^3o^3s^3 $$ |
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