Question

Z1=2(cos30°+i sin30°) Z2=1/2(cos10°+i sin10°) Z1×z2; z1÷z2; z1^2 Help please

Answer 1

Ответ:

$z_{1}z_{2}=cos40к+isin40к$

$\frac{z_{1}}{z_{2}}=4(cos20к+isin20к)$

$z_{1} ^2=4(cos60к+isin60к)$

Пошаговое объяснение:

$z_{1}=2(cos30к+isin30к)=2e^{\frac{i\pi}{6}$

$z_{2}=\frac{1}{2} (cos10к+isin10к)=\frac{1}{2} e^{\frac{i\pi}{18}$

$z_{1}z_{2}=2e^{\frac{i\pi}{6}}\frac{1}{2} e^{\frac{i\pi}{18}}=e^{\frac{i\pi}{6}+\frac{i\pi}{18}}=e^{\frac{2i\pi}{9}}=cos40к+isin40к$

$\frac{z_{1}}{z_{2}}=\frac{2e^{\frac{i\pi}{6}}}{\frac{1}{2} e^{\frac{i\pi}{18}}} =4e^{\frac{i\pi}{6}-\frac{i\pi}{18}}=4e^{\frac{i\pi}{9}}=4(cos20к+isin20к)$

$z_{1} ^2=(2e^{\frac{i\pi}{6}})^2=4e^{\frac{i\pi}{3}}=4(cos60к+isin60к)$

Answer 2

Sure, I'd be happy to help you with those complex number operations involving $Z_1$ and $Z_2$:

Given: $Z_1 = 2 \left( \cos 30° + i \sin 30° \right)$ $Z_2 = \frac{1}{2} \left( \cos 10° + i \sin 10° \right)$

Let's start by performing the requested operations:

1. Multiplication $Z_1 \times Z_2$: $Z_1 \times Z_2 = 2 \left( \cos 30° + i \sin 30° \right) \times \frac{1}{2} \left( \cos 10° + i \sin 10° \right)$

   Using the properties of complex number multiplication: $Z_1 \times Z_2 = \frac{1}{2} \left( 2 \cos 30° \cos 10° - 2 \sin 30° \sin 10° \right) + i \left( 2 \cos 30° \sin 10° + 2 \sin 30° \cos 10° \right)$

   Now, we can apply trigonometric identities: $Z_1 \times Z_2 = \frac{1}{2} \left( \cos (30° + 10°) \right) + i \left( \sin (30° + 10°) \right)$

   $Z_1 \times Z_2 = \frac{1}{2} \left( \cos 40° \right) + i \left( \sin 40° \right)$

2. Division $Z_1 \div Z_2$: $Z_1 \div Z_2 = \frac{2 \left( \cos 30° + i \sin 30° \right)}{\frac{1}{2} \left( \cos 10° + i \sin 10° \right)}$

   To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator: $Z_1 \div Z_2 = \frac{2 \left( \cos 30° + i \sin 30° \right) \left( \cos 10° - i \sin 10° \right)}{\frac{1}{2} \left( \cos 10° + i \sin 10° \right) \left( \cos 10° - i \sin 10° \right)}$

   Simplify the expression: $Z_1 \div Z_2 = \frac{2 \cos 30° \cos 10° + 2 i \cos 30° \sin 10° + 2 i \sin 30° \cos 10° - 2 \sin 30° \sin 10°}{\frac{1}{4} \left( \cos^2 10° + \sin^2 10° \right)}$

   $Z_1 \div Z_2 = \frac{2 \cos (30° + 10°) + 2 i \sin (30° + 10°)}{\frac{1}{4}}$

   $Z_1 \div Z_2 = 8 \left( \cos 40° + i \sin 40° \right)$