91Ó°ÊÓ

In this exercise, we are given a complex number in polar form and need to convert it to rectangular form. The polar form of a complex number is expressed as \text{ \( r \text { cis } \theta \) }, where \( r \) is the magnitude and \( \theta \) is the angle or argument. We need to convert this to rectangular form, which takes the form \( a + bi \), where \( a \) and \( b \) are real numbers. To do this, we use the conversion formula: \text { \[ z = r(\cos\theta + i\sin\theta) \] }. Treat \( r \) as the distance from the origin to the point, \( \theta \) as the angle in a counter-clockwise direction from the positive x-axis, and use trigonometric functions to find the real and imaginary parts.

The magnitude (or modulus) of a complex number tells us how far the number is from the origin in the complex plane. For the given problem, the magnitude \( r \) is 6. The angle (or argument) \( \theta \) shows the direction of the number. In the polar form \( 6 \text { cis } 135^\text{\textdegree} \), the angle is given as 135 degrees. To better work with it, we convert it to radians. The formula for converting from degrees to radians is: \text { \[ \text{radians} = \frac{\text{degrees} \times \text{\textpi}}{180} \] }\text { So, \( 135^\text{\textdegree} \) in radians is } \text { \[ \frac{135 \times \text{\textpi}}{180} = \frac{3\text{\textpi}}{4} \] }.

Trigonometric functions are essential to convert complex numbers from polar to rectangular form. Specifically, we use cosine (\( \text{cos} \)) and sine (\( \text{sin} \)). Let's calculate these for \( \theta = \frac{3\text{\textpi}}{4} \). \text { \( \cos(\frac{3\text{\textpi}}{4}) = -\frac{1}{\text{\textsqrt{2}}} \) } and \text { \( \sin(\frac{3\text{\textpi}}{4}) = \frac{1}{\text{\textsqrt{2}}} \). Combining these values gives us the real and imaginary parts as follows: \text { \( 6 \times \cos(\frac{3\text{\textpi}}{4}) = 6 \times -\frac{1}{\text{\textsqrt{2}}} = -3\text{\textsqrt{2}} \) } for the real part, and \text { \( 6 \times \sin(\frac{3\text{\textpi}}{4}) = 6 \times \frac{1}{\text{\textsqrt{2}}} = 3\text{\textsqrt{2}}i \) } for the imaginary part. Finally, we combine these to write the complex number in rectangular form as: \