Evaluating the Trigonometric Functions

Oct 13, 2025 by windailys.com 52 views

Introduction to De Moivre's Theorem

De Moivre's Theorem is a fundamental concept in mathematics that deals with the representation of complex numbers in polar form. It states that for any complex number written in polar form as $r(\cos \theta + i \sin \theta)$ , the $n$ th power of this number can be expressed as $r^n(\cos n\theta + i \sin n\theta)$ . In this article, we will explore the application of De Moivre's Theorem in solving a complex division problem.

Understanding the Problem

The problem requires us to divide the complex number $2 \sqrt{5}\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)$ by the complex number $\frac{1}{\sqrt{5}}\left(\cos \left(-60^{\circ}\right)+i \sin \left(-60^{\circ}\right)\right)$ . To solve this problem, we need to apply the concept of division of complex numbers in polar form.

Applying De Moivre's Theorem

To divide two complex numbers in polar form, we can use the following formula:

\frac{r_1(\cos \theta_1 + i \sin \theta_1)}{r_2(\cos \theta_2 + i \sin \theta_2)} = \frac{r_1}{r_2}(\cos (\theta_1 - \theta_2) + i \sin (\theta_1 - \theta_2))

In our problem, we have:

r_1 = 2 \sqrt{5}, \theta_1 = 60^{\circ}

r_2 = \frac{1}{\sqrt{5}}, \theta_2 = -60^{\circ}

Substituting these values into the formula, we get:

\frac{2 \sqrt{5}\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)}{\frac{1}{\sqrt{5}}\left(\cos \left(-60^{\circ}\right)+i \sin \left(-60^{\circ}\right)\right)} = \frac{2 \sqrt{5}}{\frac{1}{\sqrt{5}}}(\cos (60^{\circ} - (-60^{\circ})) + i \sin (60^{\circ} - (-60^{\circ})))

Simplifying the expression, we get:

= 2 \sqrt{5} \cdot \sqrt{5}(\cos 120^{\circ} + i \sin 120^{\circ})

= 10(\cos 120^{\circ} + i \sin 120^{\circ})

Evaluating the Trigonometric Functions

To evaluate the trigonometric functions, we need to use the following values:

\cos 120^{\circ} = -\frac{1}{2}

\sin 120^{\circ} = \frac{\sqrt{3}}{2}

Substituting these values into the expression, we get:

= 10\left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)

Simplifying the expression, we get:

= -5 + 5i\sqrt{3}

Conclusion

In this article, we have applied De Moivre's Theorem to solve a complex division problem. We have used the formula for division of complex numbers in polar form and evaluated the trigonometric functions to obtain the final result. The final answer is:

-5 + 5i\sqrt{3}

Practice Problems

To practice solving complex division problems using De Moivre's Theorem, try the following problems:

Divide the complex number $3 \sqrt{2}\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)$ by the complex number $\frac{1}{\sqrt{2}}\left(\cos \left(-45^{\circ}\right)+i \sin \left(-45^{\circ}\right)\right)$ .
Divide the complex number $4 \sqrt{3}\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)$ by the complex number $\frac{1}{\sqrt{3}}\left(\cos \left(-30^{\circ}\right)+i \sin \left(-30^{\circ}\right)\right)$ .

Solutions

To check your answers, refer to the following solutions:

$\frac{3 \sqrt{2}\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)}{\frac{1}{\sqrt{2}}\left(\cos \left(-45^{\circ}\right)+i \sin \left(-45^{\circ}\right)\right)} = 6(\cos 90^{\circ} + i \sin 90^{\circ})$

= 6i

$\frac{4 \sqrt{3}\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)}{\frac{1}{\sqrt{3}}\left(\cos \left(-30^{\circ}\right)+i \sin \left(-30^{\circ}\right)\right)} = 4 \sqrt{3} \cdot \sqrt{3}(\cos 60^{\circ} + i \sin 60^{\circ})$

= 12(\cos 60^{\circ} + i \sin 60^{\circ})

= 12\left(\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)

= 6 + 6i\sqrt{3}$$<br/> **Linear Algebra Online Trial Quiz: De Moivre's Theorem and Complex Numbers Q&A** ==================================================================== **Q: What is De Moivre's Theorem?** ------------------------------ A: De Moivre's Theorem is a fundamental concept in mathematics that deals with the representation of complex numbers in polar form. It states that for any complex number written in polar form as $r(\cos \theta + i \sin \theta)$, the $n$th power of this number can be expressed as $r^n(\cos n\theta + i \sin n\theta)$. **Q: How do I apply De Moivre's Theorem to solve complex division problems?** ------------------------------------------------------------------- A: To apply De Moivre's Theorem to solve complex division problems, you can use the following formula: $$\frac{r_1(\cos \theta_1 + i \sin \theta_1)}{r_2(\cos \theta_2 + i \sin \theta_2)} = \frac{r_1}{r_2}(\cos (\theta_1 - \theta_2) + i \sin (\theta_1 - \theta_2))

Q: What are the steps to follow when applying De Moivre's Theorem?

A: The steps to follow when applying De Moivre's Theorem are:

Write the complex numbers in polar form.
Identify the values of $r_1$ , $\theta_1$ , $r_2$ , and $\theta_2$ .
Substitute these values into the formula.
Simplify the expression.
Evaluate the trigonometric functions.

Q: How do I evaluate the trigonometric functions in De Moivre's Theorem?

A: To evaluate the trigonometric functions in De Moivre's Theorem, you need to use the following values:

$\cos \theta = \cos \theta$
$\sin \theta = \sin \theta$

You can also use the following identities:

$\cos (A + B) = \cos A \cos B - \sin A \sin B$
$\sin (A + B) = \sin A \cos B + \cos A \sin B$

Q: What are some common mistakes to avoid when applying De Moivre's Theorem?

A: Some common mistakes to avoid when applying De Moivre's Theorem are:

Not writing the complex numbers in polar form.
Not identifying the values of $r_1$ , $\theta_1$ , $r_2$ , and $\theta_2$ .
Not substituting the values into the formula correctly.
Not simplifying the expression correctly.
Not evaluating the trigonometric functions correctly.

Q: How can I practice solving complex division problems using De Moivre's Theorem?

A: You can practice solving complex division problems using De Moivre's Theorem by trying the following problems:

Divide the complex number $3 \sqrt{2}\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)$ by the complex number $\frac{1}{\sqrt{2}}\left(\cos \left(-45^{\circ}\right)+i \sin \left(-45^{\circ}\right)\right)$ .
Divide the complex number $4 \sqrt{3}\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)$ by the complex number $\frac{1}{\sqrt{3}}\left(\cos \left(-30^{\circ}\right)+i \sin \left(-30^{\circ}\right)\right)$ .

Q: What are some real-world applications of De Moivre's Theorem?

A: De Moivre's Theorem has many real-world applications, including:

Electrical engineering: De Moivre's Theorem is used to analyze and design electrical circuits.
Signal processing: De Moivre's Theorem is used to analyze and process signals.
Navigation: De Moivre's Theorem is used to calculate distances and directions.
Computer graphics: De Moivre's Theorem is used to create 3D models and animations.

Q: How can I learn more about De Moivre's Theorem and complex numbers?

A: You can learn more about De Moivre's Theorem and complex numbers by:

Reading books and online resources.
Watching video tutorials and lectures.
Practicing problems and exercises.
Joining online communities and forums.
Taking online courses and classes.