Introduction to De Moivre's Theorem Calculator:
De Moivre's Theorem calculator is an online tool that is used to find the de Moivre`s theorem problem in a few seconds. Our tool simplifies the process of complex numbers and their powers in finding their roots in calculations.
The demoivre's theorem calculator is the best online source that helps you easily get solutions of nth roots in nth power of complex numbers quickly because sometimes these complex number problems become complicated and not easily solve in manual evaluation.
What is De Moivre Threoem?
De Moivre's Theorem is a process that is used to find the power of complex numbers and find their roots. It is a fundamental theorem in complex number theory. It is a straightforward way to convert polar coordinates into rectangles.
De Moivre's Theorem states that a polar coordinates z is equal to the magnitude r and rectangular coordinates in n-th power of a complex number. It is given as,
$$ z^n \;=\; \biggr(r^n \times (cos(n \theta) + \iota \times sin(n \theta))) $$
Where:
- z=r(cosθ+isinθ) is a complex number in polar form.
- r is the magnitude (or modulus) of z as |z|=r
- θ is the argument of complex number z.
- n is an integer (that may be positive, negative, or zero).
Working Method of Demoivre's Theorem Calculator:
De moivre theorem calculator has a simple working method for calculating the complex number powers and their roots without any error.
You can add any type of complex number problem in this de moivre's theorem calculator, it will give you solutions according to the rules of the demovire theorem. Let's suppose an example that helps you to understand the calculation process.
Example: Find the power of complex numbers 3:
$$ z \;=\; 1 + \iota $$
Solution:
Step 1:
Convert the complex number into a polar form such as z=x+yi. Then the given question becomes
$$ z \;=\; z,\; x \;=\; 1\; and,\; y \;=\; 1\; (i\; coefficient\; is\; 1) $$
Step 2:
Find the magnitude r as r=|z|.
$$ r \;=\; |z| \;=\; \sqrt{x^2 + y^2} \;=\; \sqrt{1^2 + 1^2} \;=\; \sqrt{2} $$
Step 3:
Now it finds the argument angle θ.
$$ \theta \;=\; tan^{-1} \biggr( \frac{y}{x} \biggr) \;=\; tan^{-1}(1) \;=\; \frac{\pi}{4} $$
Step 4:
You have r and argument angle θ value so the complex number is written as,
$$ z \;=\; r(cos\; \theta + \iota sin\; \theta) \;=\; \sqrt{2}(cos \frac{\pi}{4} + \iota sin \frac{\pi}{4} ) $$
Step 5:
Apply the De Moivre theorem as:
$$ (r(cos \theta + \iota sin \theta))^n \;=\; r^n (cos(n \theta) + \iota sin(n\; \theta)) $$
Step 6:
Add all the values in the De Moivre theorem and simplify the given expression.
$$ (\sqrt{2}(cos \frac{\pi}{4} + \iota sin \frac{\pi}{4}))^3 \;=\; (\sqrt{2})^3 (cos (3 . \frac{\pi}{4}) + \iota sin (3 . \frac{\pi}{4} )) $$
Step 7:
After simplification, we get the solution of finding the power of complex numbers is,
$$ 2\sqrt{2} (cos \frac{3 \pi}{4} + \iota sin \frac{3 \pi}{4}) \;=\; 2\sqrt{2} \biggr( -\frac{1}{\sqrt{2}} + \iota \frac{1}{\sqrt{2}} \biggr) \;=\; -2 + 2\iota $$
Example: To find the roots of a complex number:
$$ z \;=\; 1 + \iota $$
Solution:
Step 1:
Convert the given example into polar form. As
$$ z \;=\; z,\; x \;=\; 1,\; y \;=\; 1 $$
$$ z \;=\; x + yi $$
Step 2:
Find the magnitude r and argument angle θ using the above values.
$$ r \;=\; |z| \;=\; \sqrt{x^2 + y^2} \;=\; \sqrt{1^2 + 1^2} \;=\; \sqrt{2} $$
$$ \theta \;=\; tan^{-1} \biggr(\frac{y}{x} \biggr) \;=\; tan^{-1}(1) \;=\; \frac{\pi}{4} $$
Step 3:
Express the above polar form values into a complex number,
$$ z \;=\; r(cos \theta + \iota sin \theta) \;=\; \sqrt{2} (cos \frac{\pi}{4} + \iota sin \frac{\pi}{4} ) $$
Step 4:
The De Moivre threoem for finding complex number root is,
$$ z_k \;=\; r^{\frac{1}{n}} \biggr(cos \biggr(\frac{\theta + 2 k \pi}{n} \biggr) + \iota sin \biggr( \frac{\theta + 2 k \pi}{n} \biggr) \biggr) $$
$$ k \;=\; 0,1,2,3,...,n-1 $$
Step 5:
Put all the values in the above De Moivre theorem formula,
$$ z_k \;=\; (\sqrt{2})^{\frac{1}{3}} \biggr(cos \biggr(\frac{\frac{\pi}{4} + 2 k \pi}{3} \biggr) + \iota sin \biggr( \frac{\frac{\pi}{4} + 2 k \pi}{3} \biggr) \biggr) $$
Step 6:
As you find the cube root so k=0,1,2, calculate the roots for k=0,1,2,
For k = 0:
$$ z_0 \;=\; 2^{\frac{1}{6}} \biggr(cos \frac{\pi}{12} + \iota sin \frac{\pi}{12} \biggr) $$
For k = 1:
$$ z_1 \;=\; 2^{\frac{1}{6}} \biggr(cos \frac{9 \pi}{12} + \iota sin \frac{9 \pi}{12} \biggr) $$
For k = 2:
$$ z_2 \;=\; 2^{\frac{1}{6}} \biggr(cos \frac{17 \pi}{12} + \iota sin \frac{17 \pi}{12} \biggr) $$
How to Use the De Moivre's Theorem Calculator?
The de moivre theorem calculator has a simple design that makes it easy for you to know how to use it for the evaluation of complex number problems, only when you follow our guidelines which are given as:
- Enter the expression of the complex number in the given input field.
- Enter the number of root of your complex number through which you want to find the roots of the complex number in the input field.
- Recheck the given complex number expression before clicking the calculate button to start the evaluation process in demoivre's theorem calculator.
- Click the “Calculate” button to get the result of your given complex number problem.
- If you are trying our de moivre calculator for the first time then you can use the load example to learn more about this concept.
- Click on the “Recalculate” button to get a new page for finding more example solutions to complex number problems.
Final Result of De Moivre Theorem Calculator:
De moivre's theorem calculator gives you the solution from a given complex number when you add the input into it. It included as:
- Result Option:
When you click on the result option, it gives you a solution to the complex number to find its roots and nth power.
- Possible Steps:
When you click on it, this option will provide you with a solution where all the calculations of the De Moivre Theorem process are mentioned.
Benefits of Using De Moivre Calculator:
The demoivre's theorem calculator provides you with tons of benefits that help you to calculate the power and root of complex number problems and give you solutions without any trouble. These benefits are:
- The de moivre theorem calculator is a free-of-cost tool so you can use it for free to find complex number problem solutions without spending.
- It is a manageable tool that can manage various types of complex number problems because it has advanced features.
- It gives you conceptual clarity for the demoivre theorem process when you use it for practice to solve more examples.
- It saves the time that you consume on the calculation of complex number problems manually.
- It is a reliable tool that provides you with accurate solutions whenever you use it to calculate complex numbers without any man-made mistakes in calculation.
- De moivre's theorem calculator allows you to use it multiple times for the evaluation of complex number problem.
