Calculate these complex numbers and express your results in...

Calculate these complex numbers and express your results in rectangular form:
(a) $\frac{15 \angle 45^{\circ}}{3-j 4}+j 2$
(b) $\frac{8 \angle-20^{\circ}}{(2+j)(3-j 4)}+\frac{10}{-5+j 12}$
(c) $10+\left(8 \angle 50^{\circ}\right)(5-j 12)$

Key Concepts

Polar Form

Polar form represents a complex number by its magnitude (or modulus) and its angle (or phase), written as r ? ?. This form simplifies multiplication, division, and exponentiation because the operations can be carried out by manipulating magnitudes and angles directly.

Conversion between Forms

Conversion between polar and rectangular forms is a vital skill in complex number arithmetic. To convert from polar to rectangular, the formulas a = r cos ? and b = r sin ? are used. Conversely, to go from rectangular to polar, one computes the magnitude r = ?(a² + b²) and determines the angle ? = arctan(b/a).

Complex Numbers

Complex numbers comprise a real and an imaginary part and can be expressed in various forms. The two most common representations are the rectangular form (a + bj) and the polar form (r ? ?). Understanding these representations is essential for performing various operations and conversions between different forms.

Rectangular Form

Rectangular form expresses a complex number as a sum of its real and imaginary components, denoted as a + bj. This form is especially useful for addition and subtraction of complex numbers, as the operations are performed component-wise.

Arithmetic Operations with Complex Numbers

Performing operations such as addition, subtraction, multiplication, and division on complex numbers often requires using both rectangular and polar forms depending on which simplifies the computation. Multiplication and division tend to be more straightforward in polar form, while addition and subtraction are easier in rectangular form. Understanding how and when to switch between these forms is crucial for efficient and accurate problem solving.

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