Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Write each complex number in the trigonometric form \(r(\cos \theta+i \sin \theta),\) where \(r\) is exact and \(0^{\circ} \leq \theta<360^{\circ}\) $$1+i \sqrt{3}$$

Step-by-Step Solution

Answer

The trigonometric form is \( 2(\cos 60^{\circ} + i \sin 60^{\circ}) \).

Step 1: Find the Magnitude

For a complex number in the form \( a + bi \), the magnitude \( r \) is calculated using the formula \( r = \sqrt{a^2 + b^2} \). Here, \( a = 1 \) and \( b = \sqrt{3} \). Substitute these values: \( r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \).

Step 2: Determine the Angle

The angle \( \theta \) can be found using the formula \( \tan \theta = \frac{b}{a} \). So, \( \tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3} \). The angle for which \( \tan \theta = \sqrt{3} \) is \( \theta = 60^{\circ} \).

Step 3: Write the Trigonometric Form

Now that we have \( r = 2 \) and \( \theta = 60^{\circ} \), the complex number \( 1 + i \sqrt{3} \) in trigonometric form is \( 2(\cos 60^{\circ} + i \sin 60^{\circ}) \).

Key Concepts

Complex Numbers

Complex numbers are a significant part of mathematics, where they extend the idea of one-dimensional number lines into two dimensions. They are of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined by the equation \(i^2 = -1\).

• Real Part: This is the "\(a\)" in \(a + bi\), representing the horizontal axis on a complex plane.
• Imaginary Part: The "\(b\)" value, which pairs with the imaginary unit "\(i\)", showing the vertical positioning.

Together, these components allow complex numbers to represent points or vectors in a two-dimensional space, significant for calculations in fields like engineering, physics, and more. Hence, a complex number is not just two numbers but a pair that describes direction and magnitude.

Magnitude of Complex Numbers

The magnitude, often referred to as the modulus, of a complex number intuitively represents its distance from the origin (0,0) on the complex plane. For a complex number \(a + bi\), we calculate its magnitude \(r\) using the formula \[ r = \sqrt{a^2 + b^2}\]This formula derives from the Pythagorean theorem, recognizing that every complex number maps onto a right triangle's hypotenuse.

For example, if your complex number is \(1 + i \sqrt{3}\), then the calculations would unfold as follows:

• Identify: \(a = 1\), \(b = \sqrt{3}\)
• Substitute: \(r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2\)

Magnitude gives us not just the distance but helps specify the size without direction, critical when moving towards polar forms.

Angle of Complex Numbers

To fully express a complex number in its trigonometric form, we require not just its magnitude but also an angle that encapsulates its direction relative to the positive real axis. This angle, known as \(\theta\), can be determined by the formula \[ \tan \theta = \frac{b}{a}\]Using this relationship, if we have a complex number \(1+i\sqrt{3}\), we calculate:

• Identify: \(a = 1\), \(b = \sqrt{3}\)
• Calculate: \(\tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3}\)
• Recognize: An angle \(\theta\) satisfying \(\tan \theta = \sqrt{3}\) is \(60^{\circ}\).

Understanding the angle is crucial because it defines the complex number’s rotational positioning, an aspect easy to overlook while focusing purely on magnitudes. It’s necessary when translating to polar coordinate forms, as it ensures that directions are accurately conveyed.

Polar Coordinates

Polar coordinates offer a new lens through which to view complex numbers, transitioning from Cartesian's \((x, y)\) coordinates to a system denoted by a radius and angle \((r, \theta)\). This form aligns with vectors, making it useful in fields like physics and engineering for representing rotational dynamics.

In polar form, a complex number's components are depicted as:

• Magnitude (\(r\)): The length from the origin, or "how far."
• Angle (\(\theta\)): The direction or "which way," oriented counter-clockwise from the positive x-axis.

The conversion into a trigonometric or polar form \(r(\cos \theta + i\sin \theta)\) does this transformation, giving us a concise way to compile both radial and angular data. For instance, with \(1+i\sqrt{3}\), this turns into \(2(\cos 60^{\circ} + i\sin 60^{\circ})\). This format grants greater insight when geometric interpretations and simplifications are needed.