If \([(1+2 i) /(2+i)]=r(\cos \theta+i \sin \theta)\), then (a) \(r=1, \theta=\tan ^{-1}(4 / 3)\) (b) \(\mathrm{r}=\sqrt{5}, \theta=\tan ^{-1}(5 / 4)\) (c) \(\mathrm{r}=1, \theta=\tan ^{-1}(3 / 4)\) (d) \(r=2, \theta=\tan ^{-1}(3 / 4)\)

Understanding how to work with complex numbers often involves rationalizing the denominator, especially when division is involved. This process allows for a clearer representation of complex numbers and simplifies further operations. Rationalizing involves the multiplication of the numerator and the denominator of the complex fraction by the conjugate of the denominator. The conjugate of a complex number is simply the number with the same real part and the opposite imaginary part. For example, the conjugate of \(2+i\) is \(2-i\). When we multiply a complex number by its conjugate, the result is always a real number, specifically the square of the magnitude of the complex number. In the case of \((2+i)(2-i)\), the product is \(4+1=5\), which is a real number and eliminates the complex part in the denominator.

This step is vital as it transforms the complex number into a more usable form without changing its value, which profoundly impacts our subsequent calculations, such as when converting complex numbers to their polar form.

Converting complex numbers into polar form is a crucial skill in understanding and manipulating these numbers, especially in fields such as engineering and physics, where the magnitude and direction of vectors are often analyzed. Polar form expresses a complex number in terms of its magnitude \(r\) and its angle \(\theta\), which is also known as the argument of the complex number. To find \(r\), we use the Pythagorean theorem, calculating the square root of the sum of the squares of the real and imaginary parts. For the complex number \(4/5 + 3/5i\), the magnitude is \(\sqrt{(4/5)^2 + (3/5)^2} = 1\).

To find the argument \(\theta\), we measure the angle that the line, representing the complex number, makes with the positive real axis in the complex plane. This is found using the inverse tangent function, where the imaginary part is divided by the real part \(\tan^{-1}(\frac{\text{Im}(z)}{\text{Re}(z)})\). For \(4/5 + 3/5i\), the argument is \(\tan^{-1}(\frac{3}{4})\). These values, \(r\) and \(\theta\), together represent the complex number in polar form: \(r(\cos \theta + i \sin \theta)\).

The Pythagorean theorem is a fundamental principle in mathematics that extends its applicability to complex numbers when determining their magnitude. While the theorem is traditionally associated with right-angled triangles, it applies to the complex plane by treating the real and imaginary components of a complex number as the two perpendicular sides of a triangle. The magnitude, or modulus, of the complex number represents the hypotenuse of this triangle.

For example, if we have a complex number \(a+bi\), the length of the 'real' side is \(a\), and the length of the 'imaginary' side is \(b\). The magnitude \(r\) is then \(\sqrt{a^2 + b^2}\), which corresponds to the hypotenuse of a triangle with sides of lengths \(a\) and \(b\). This relationship is key to converting complex numbers from their standard form to polar form and helps illustrate the geometric interpretation of complex numbers.

The argument of a complex number is essentially the direction of the number when represented as a point in the complex plane, which is a two-dimensional plane where the x-axis represents the real part and the y-axis the imaginary part of the number. The argument is the angle the line makes with the positive x-axis, measured counterclockwise from the x-axis to the line segment connecting the origin to the point.

When calculating the argument \(\theta\), we use trigonometric functions such as the arctangent function, as seen in the inverse tangent \(\tan^{-1}\), to find this angle. It's important to note that the inverse tangent function can only determine the angle to an accuracy of \(180^\circ\), so we have to determine the quadrant in which the complex number lies to find the exact angle. The argument is usually given in radians and is essential in expressing complex numbers in their polar form, allowing for easier multiplication, division, and finding powers and roots of complex numbers.