De Moivre's formula

In mathematics, de Moivre's formula is a special rule that helps us work with complex numbers. It says that for any real number x and integer n, we can raise the combination of cosine x plus i times sine x to the power of n, and it will equal cosine of n times x plus i times sine of n times x. Here, i is the imaginary unit, which is a special number where i squared equals negative one. This formula is named after Abraham de Moivre, even though he did not actually write it down himself.

This formula is important because it links together complex numbers and trigonometry. By using algebra to expand the left side, we can find new ways to express cosine and sine of many multiples of an angle using only the cosine and sine of the original angle. The expression cos x plus i sin x is sometimes shortened to cis x, making the formula easier to read.

While the formula works perfectly for whole number powers, there are ways to extend it to work with other kinds of exponents too. These extensions help us understand special complex numbers called roots of unity, which are numbers that, when raised to a power n, equal one. With some extra work, the formula can even be used when x is not a simple real number, but a more general complex number.

Example

De Moivre's formula helps us work with special numbers called complex numbers. For example, if we take a number like (\cos(\pi/6) + i\sin(\pi/6)) and square it (multiply it by itself), the formula tells us the result will be (\cos(2 \cdot \pi/6) + i\sin(2 \cdot \pi/6)).

In simpler terms, this formula shows how to raise these special numbers to any power and still get a correct result using cosine and sine functions. It’s a useful tool in advanced math!

Relation to Euler's formula

De Moivre's formula is closely related to Euler's formula, which connects trigonometric functions with the complex exponential function. Euler's formula states that eⁱx equals cos x + i sin x, where x is measured in radians instead of degrees.

Using Euler's formula and the exponential law for integer powers, we can derive de Moivre's formula. This shows that raising (cos x + i sin x) to the power of n gives the same result as cos(nx) + i sin(nx). This relationship helps us understand how trigonometry and complex numbers work together.

Proof by induction

De Moivre's theorem can be shown to be true using a method called mathematical induction. This method helps us prove that something works for all whole numbers by checking it step by step.

We start by checking that the theorem works for the number 1. It does! Then we assume it works for some number k, and we show that if it works for k, it must also work for k + 1. By doing this, we prove it works for all positive whole numbers. We also find that it works for zero and negative whole numbers too, completing the proof.

See angle sum and difference identities.

Formulae for cosine and sine individually

See also: List of trigonometric identities

These formulas show how to find the sine and cosine of numbers that are multiples of other numbers, using simple steps. For example, we can find the cosine of twice an angle (2x) or three times an angle (3x) by using the cosine and sine of the original angle (x).

The formulas help us calculate things like cos 2x or sin 2x by using the values of cos x and sin x. They are useful in many areas of math and science.

Failure for non-integer powers, and generalization

De Moivre's formula works well when we raise a complex number to an integer power, but it doesn't work the same way for non-integer powers. When we try to raise a complex number to a non-integer power, we get many possible answers instead of just one.

We can use a special version of de Moivre's formula to find the n-th roots of a complex number. If a complex number is written in a certain form, we can find its roots by adjusting the angle in a specific way.

For more general powers, things become even more complex, and we end up with many possible answers depending on a special number k. If the power is a simple fraction, we get a few answers, but for other powers, there are infinitely many possible results.

Analogues in other settings

De Moivre's formula can also be used in different mathematical areas. In hyperbolic trigonometry, a similar rule works with special functions called "hyperbolic cosine" (cosh) and "hyperbolic sine" (sinh). For whole numbers n, the formula says that when you raise the sum of cosh x and sinh x to the power n, you get cosh of n times x plus sinh of n times x.

The formula also extends to complex numbers and even to structures called quaternions and special 2×2 matrices, showing how widely this important idea applies in mathematics.