Problem 10 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

\(-1+\sqrt{-3}=r e^{\theta}\), then \(\theta=\) (a) \(\frac{2 \pi}{3}\) (b) \(-\frac{2 \pi}{3}\) (c) \(\frac{\pi}{3}\) (d) \(-\frac{\pi}{3}\)

Step-by-Step Solution

Answer

The correct answer is \(\theta = \frac{2\pi}{3}\), option (a).

Step 1: Convert to Complex Form

First, express the given expression \( -1 + \sqrt{-3} \) in standard form for complex numbers. Notice \( \sqrt{-3} = \sqrt{3}i \), so the complex number is \( -1 + \sqrt{3}i. \)

Step 2: Determine the Magnitude

To find \(r\), which is the magnitude (modulus) of the complex number, use the formula \[ r = \sqrt{x^2 + y^2} \] where \(x = -1\) and \(y = \sqrt{3}\). This gives:\[ r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2. \]

Step 3: Find the Argument

The argument \(\theta\) of the complex number is determined using \[ \theta = \tan^{-1}\left( \frac{y}{x} \right) \] Substitute \(x = -1\) and \(y = \sqrt{3}\) into the equation:\[ \theta = \tan^{-1}\left( \frac{\sqrt{3}}{-1} \right) = \tan^{-1}(-\sqrt{3}) \]This corresponds to angles of \(-\frac{\pi}{3}\) or \(\frac{2\pi}{3}\) (in the second quadrant where sine is positive and tangent is negative).

Step 4: Select the Correct Argument angle

Thoroughly evaluate where the complex number is located in the complex plane. The real part is negative and the imaginary part is positive, placing it in the second quadrant of the complex plane. This aligns with the positive argument \(\theta = \frac{2\pi}{3}\).

Step 5: Encode Result

Thus, the angle \(\theta = \frac{2\pi}{3}\). From the choices provided, this matches option (a).

Key Concepts

Magnitude of Complex Numbers

The magnitude of a complex number, often denoted as the modulus, measures how far the number is from the origin on the complex plane. For a complex number expressed in the form \(z = a + bi\), the real part is \(a\) and the imaginary part is \(bi\). To find the magnitude, or modulus, we use the formula:\[ r = \sqrt{a^2 + b^2} \]This formula is similar to the Pythagorean theorem and represents a distance.

• The real part (\(a\)) serves as the horizontal component (x-axis).
• The imaginary part (\(b\)) serves as the vertical component (y-axis).

By squaring each part, adding the results, and taking the square root, you determine the distance from the origin (\(0,0\)) to the point (\(a,b\)) on the plane. In the problem of \(-1 + \sqrt{3}i\), our values are \(a = -1\) and \(b = \sqrt{3}\), leading to:\[ r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]So, the magnitude of the complex number is 2.

Argument of Complex Numbers

The argument of a complex number, denoted as \(\theta\), is the angle that the line representing the complex number makes with the positive real axis in the complex plane. It's like describing the direction of our complex number.The formula used to find \(\theta\) is:\[ \theta = \tan^{-1}\left( \frac{b}{a} \right) \]where:

• \(b\) is the imaginary part.
• \(a\) is the real part.

Importantly, knowing the quadrant where the complex number is, makes it easier to interpret \(\theta\). For our example of \(-1 + \sqrt{3}i\), which is located in the second quadrant (since the real part is negative and the imaginary part is positive), the calculations for \(\theta\) become:\[ \theta = \tan^{-1}\left( \frac{\sqrt{3}}{-1} \right) = \tan^{-1}(-\sqrt{3}) \]Given the quadrant and standard angles, the appropriate argument is \(\frac{2\pi}{3}\), aligning with the direction where the vector points in the second quadrant.

Polar Form of Complex Numbers

Complex numbers can also be expressed in polar form, bridging the gap between algebra and geometry. Polar form combines the magnitude and argument:\[ z = r \left( \cos(\theta) + i \sin(\theta) \right) \]Alternatively, this can be neatly written using Euler’s formula:\[ z = re^{i\theta} \]where:

• \(r\) is the magnitude.
• \(\theta\) is the argument, both calculated previously.

Given our complex number, the polar form becomes:\[ z = 2e^{i \frac{2\pi}{3}} \]This not only tells us how far the number is from the origin, but also the angle at which it lies. Converting complex numbers to polar form can simplify performing operations like multiplication and division, due to its concise representation of both magnitude and direction.