Complex numbers can be expressed in polar form, which simplifies operations like multiplication. When multiplying two complex numbers in polar form, the process is straightforward: multiply the magnitudes (r values) and add the angles (θ values). This method avoids the need to convert to rectangular form, making calculations more efficient.
For example, to find the product of two complex numbers represented as \( r_1 \text{cis} \theta_1 \) and \( r_2 \text{cis} \theta_2 \), you would calculate:
\[ r_1 \cdot r_2 \text{cis} (\theta_1 + \theta_2) \]
Consider two complex numbers: \( 3 \text{cis} 15^\circ \) and \( 2 \text{cis} 30^\circ \). To multiply them, first multiply the r values:
\[ 3 \times 2 = 6 \]
Next, add the angles:
\[ 15^\circ + 30^\circ = 45^\circ \]
Thus, the product is:
\[ 6 \text{cis} 45^\circ \]
This notation, where \( \text{cis} \) stands for \( \cos \theta + i \sin \theta \), provides a compact way to express complex numbers in polar form.
In another example, consider \( 4 \text{cis} \frac{\pi}{6} \) and \( 5 \text{cis} \frac{\pi}{3} \). Multiply the r values:
\[ 4 \times 5 = 20 \]
Then, add the angles:
\[ \frac{\pi}{6} + \frac{\pi}{3} \]
To add these fractions, convert \( \frac{\pi}{3} \) to have a common denominator:
\[ \frac{\pi}{3} = \frac{2\pi}{6} \]
Now, add the numerators:
\[ \frac{\pi + 2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]
Thus, the product is:
\[ 20 \text{cis} \frac{\pi}{2} \]
In summary, when multiplying complex numbers in polar form, remember to multiply the magnitudes and add the angles. Using the \( \text{cis} \) notation simplifies the expression, making it easier to work with complex numbers in various mathematical contexts.
15 Terms
