Problem 120 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[6\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)\right]^{4}$$

Step-by-Step Solution

Answer

Therefore, \[\left[6\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)\right]^{4}\] can be written in standard form as \(648 + 1122.02i\).

Step 1: Apply DeMoivre's Theorem

DeMoivre's Theorem says that when raising a complex number (in polar form) to an integer power n, we get: \[(r(\cos θ+ i \sin θ))^n = r^n(\cos nθ + i \sin nθ)\] So, applying this to our equation, we substitute r with 6, θ with 15° and n with 4: \[\left[6\left(\cos 15^{\circ}+ i \sin 15^{\circ}\right)\right]^{4} = 6^4 \left(\cos(4*15^{\circ}) + i \sin(4*15^{\circ})\right)\]

Step 2: Evaluate the values

Calculating the powers and multiplication from step 1, we get: \[1296 (\cos 60^{\circ}+ i \sin 60^{\circ})\] Now, the value of ∠60° can easily be found from the unit circle or trigonometric values, giving us: \[1296 (\frac{1}{2} + i \frac{\sqrt{3}}{2})\]

Step 3: Multiply the magnitude with the standard form numbers

By multiplying the magnitude with the standard form we get: \[1296 * \frac{1}{2}+ 1296 * i \frac{\sqrt{3}}{2} = 648 + 1122.02i\]

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary part involves the imaginary unit \(i\), which is defined as \(i^2 = -1\).
These numbers are essential in many fields of science and engineering because they allow for solutions to equations that wouldn't otherwise have real solutions.

• Real Part: Located on the horizontal axis.
• Imaginary Part: Situated on the vertical axis.

Visualizing them on a plane called the complex plane helps understand their behavior better.

Polar Form

Polar form is a way of expressing complex numbers using magnitudes and angles, rather than the traditional Cartesian form \(a + bi\). In polar form, a complex number is represented as \(r(\cos \theta + i \sin \theta)\).

• Magnitude \(r\): Distance from the origin to the point, calculated as \(\sqrt{a^2 + b^2}\).
• Angle \(\theta\): Direction of the point, often measured in degrees or radians, found using \(\tan^{-1}(b/a)\).

This form is particularly useful when multiplying or dividing complex numbers, as well as finding powers and roots. It's the key format used in DeMoivre's Theorem.

Integer Powers

Integer powers of complex numbers are easily calculated using DeMoivre's Theorem. When you have a complex number in polar form, raising it to a power becomes straightforward.
The theorem states: \([r(\cos \theta + i \sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta)\).

• Raise the magnitude \(r\) to the power \(n\).
• Multiply the angle \(\theta\) by the power \(n\).

This approach simplifies computation especially when dealing with higher powers or roots, streamlining problems into easy trigonometric calculations.

Trigonometric Values

Trigonometric values are essential in converting complex numbers back to their standard form from polar form. Key trigonometric functions like cosine and sine help bridge that transition.
These functions can be immediately determined from the unit circle, particularly for well-known angles like \(0^\circ, 30^\circ, 45^\circ, 60^\circ,\) and \(90^\circ\).

• Cosine \(\cos\): Gives the x-coordinate on the unit circle.
• Sine \(\sin\): Provides the y-coordinate on the unit circle.

When we computed \(\cos 60^\circ\) and \(\sin 60^\circ\), these values helped form the standard form: \(1296(\frac{1}{2} + i\frac{\sqrt{3}}{2})\). Understanding these values is crucial for transforming and interpreting complex numbers seamlessly.