Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 2 \sqrt{3}+2 i $$

Step-by-Step Solution

Answer

The trigonometric form is \(4(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6})\).

Step 1: Identify the real and imaginary parts

The given complex number is \(2\sqrt{3} + 2i\). Here, the real part \(a\) is \(2\sqrt{3}\) and the imaginary part \(b\) is \(2\).

Step 2: Find the magnitude of the complex number

The magnitude (or modulus) of a complex number is given by \(|z| = \sqrt{a^2 + b^2}\). Calculate it as follows:\[|z| = \sqrt{(2\sqrt{3})^2 + 2^2} = \sqrt{4 \times 3 + 4} = \sqrt{12 + 4} = \sqrt{16} = 4\]

Step 3: Determine the angle \(\theta\)

The angle \(\theta\) for the trigonometric form is found using \(\tan \theta = \frac{b}{a}\). Calculate it as follows:\[\tan \theta = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}}\]The angle \(\theta\) that satisfies this in the first quadrant is \(\theta = \frac{\pi}{6}\) since \(\tan\frac{\pi}{6} = \frac{1}{\sqrt{3}}\).

Step 4: Write the complex number in trigonometric form

The trigonometric form of a complex number is \(z = |z|(\cos\theta + i\sin\theta)\). Here, \(|z| = 4\) and \(\theta = \frac{\pi}{6}\), so we get:\[z = 4(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6})\]

Key Concepts

Trigonometric Form

The trigonometric form of a complex number is an especially neat way of expressing complex numbers in terms of their magnitude and direction. This method is particularly advantageous when dealing with multiplication, division, or powers of complex numbers.

In the trigonometric form, a complex number is expressed as:

• \( z = r(\cos \theta + i\sin \theta) \)

Here, \(r\) represents the magnitude of the complex number, and \(\theta\) is the angle, typically in radians, that the complex number makes with the positive real axis.

This format provides a geometric representation of complex numbers on the complex plane, where \(\cos \theta\) and \(\sin \theta\) represent the real and imaginary components respectively, multiplied by the magnitude \(r\).

When given a complex number like \(2\sqrt{3} + 2i\), converting it into trigonometric form allows us to handle the number more intuitively in various mathematical operations, particularly those involving rotations and scaling in the complex plane.

Magnitude of Complex Numbers

The magnitude, or modulus, of a complex number \(a + bi\) is essentially the distance from the origin to the point \((a, b)\) on the complex plane. It gives us a sense of how large the complex number is.

The formula to compute the magnitude is:

• \(|z| = \sqrt{a^2 + b^2}\)

For the given complex number, \(2\sqrt{3} + 2i\):

• We identify \(a = 2\sqrt{3}\) and \(b = 2\).
• Substituting these into the magnitude formula gives us:
• \(|z| = \sqrt{(2\sqrt{3})^2 + 2^2} = \sqrt{16} = 4\).

This means the magnitude of our complex number is 4, indicating it is 4 units away from the origin on the complex plane. Calculating the magnitude is an important step in expressing a complex number in trigonometric form as it determines the coerraform scaling factor of the entire expression.

Angle in Radians

Finding the angle \(\theta\), also known as the argument of a complex number, is vital in converting it into its trigonometric form. This angle tells us the direction of the complex number on the complex plane.

The angle is calculated using the tangent function:

• \(\tan \theta = \frac{b}{a}\)

where \(a\) is the real component and \(b\) is the imaginary component of the number. For our complex number \(2\sqrt{3} + 2i\):

• \(\tan \theta = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}}\)

The angle \(\theta\) that satisfies this condition in the first quadrant is \(\theta = \frac{\pi}{6}\), because \(\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}\).

Using radians is standard in mathematics for angles, as it relates directly to the unit circle and often simplifies the mathematics involved, especially in calculus and complex analysis. This angle is crucial as it completes the trigonometric form by informing the direction component, alongside the magnitude.