To convert the complex number from polar to rectangular form, we use the fact that a complex number in polar form is represented as:
[tex]\[ z = r \operatorname{cis}(\theta) \][/tex]
Here, [tex]\( r \)[/tex] is the magnitude (or modulus) of the complex number, and [tex]\( \theta \)[/tex] is the angle (or argument) in degrees.
The rectangular form of a complex number is given by:
[tex]\[ z = x + yi \][/tex]
where [tex]\( x \)[/tex] is the real part and [tex]\( y \)[/tex] is the imaginary part.
To convert from polar to rectangular form, we use the following relationships:
[tex]\[ x = r \cos(\theta) \][/tex]
[tex]\[ y = r \sin(\theta) \][/tex]
Given the complex number [tex]\( z = 9 \operatorname{cis}(23^\circ) \)[/tex]:
1. The magnitude [tex]\( r \)[/tex] is 9.
2. The angle [tex]\( \theta \)[/tex] is 23 degrees.
First, we need to convert the angle from degrees to radians since trigonometric functions typically use radians. The conversion from degrees to radians is done using the formula:
[tex]\[ \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \][/tex]
However, we proceed with directly applying the trigonometric functions to find the real and imaginary parts:
[tex]\[ x = 9 \cos(23^\circ) \][/tex]
[tex]\[ y = 9 \sin(23^\circ) \][/tex]
After calculating the values:
- The real part [tex]\( x \approx 8.28 \)[/tex]
- The imaginary part [tex]\( y \approx 3.52 \)[/tex]
So, the rectangular form of the complex number [tex]\( z \)[/tex] is:
[tex]\[ z \approx 8.28 + 3.52i \][/tex]
Therefore, the complex number [tex]\( z = 9 \operatorname{cis}(23^\circ) \)[/tex] in rectangular form is:
[tex]\[ z \approx 8.28 + 3.52i \][/tex]
