Leiten Sie aus der Formel von Moivre und unter Verwendung der Binomischen Formel die folgenden trigonometrischen Beziehungen her: $$ \begin{aligned} &\sin (3 \varphi)=3 \cdot \sin \varphi-4 \cdot \sin ^{3} \varphi \\ &\cos (3 \varphi)=4 \cdot \cos ^{3} \varphi-3 \cdot \cos \varphi \end{aligned} $$ Anleitung: Mit \(z=\cos \varphi+j-\sin \varphi\) lautet die Formel von Moivre: $$ (\cos \varphi+j \cdot \sin \varphi)^{n}=\cos (n \varphi)+j \cdot \sin (n \varphi) $$

De Moivre's Theorem is a powerful tool in the world of complex numbers and trigonometry. It states that for any complex number in the form \(\cos \varphi + j \cdot \sin \varphi\), raising it to a power \(n\) results in another expression involving trigonometric functions. The theorem is written as:
\[ (\cos \varphi + j \cdot \sin \varphi)^n = \cos(n \varphi) + j \cdot \sin(n \varphi) \]
This theorem helps in simplifying complex number problems, especially when dealing with powers and roots. In our exercise, it allows us to transform a trigonometric expression into a polynomial form, which can then be manipulated using other mathematical tools like the Binomial Theorem.

The Binomial Theorem provides a way to expand expressions that are raised to a power. For two terms \(a\) and \(b\) raised to an \(n\)th power, the theorem states:
\[ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \]
Here, \(\binom{n}{k}\) represents the binomial coefficients. This theorem simplifies the expansion of terms like \( (\cos \varphi + j \cdot \sin \varphi)^3 \) in our solution. Using the Binomial Theorem, we expand this expression into multiple terms, each of which can then be simplified further.

Complex numbers are an extension of the real numbers and include an imaginary unit, denoted as \(j\), where \(j^2 = -1\). Any complex number can be written in the form \( a + bj \), where \(a\) and \(b\) are real numbers. In the context of this exercise, we use the complex number representation of trigonometric functions:
\[ \cos \varphi + j \cdot \sin \varphi \]
This form helps in utilizing de Moivre's Theorem and other algebraic tools efficiently. Understanding how to manipulate complex numbers, such as simplifying powers of \(j\), is crucial in deriving trigonometric identities.

Trigonometry deals with the relationships between angles and sides of triangles, often captured through functions like sine and cosine. In our exercise, trigonometric functions are incorporated within the context of complex numbers and de Moivre's Theorem.
By expressing trigonometric identities through polynomial expansions, we reinforce how algebraic techniques intertwine with trigonometric principles. For example, we found that:
\[ \sin(3\varphi) = 3 \cdot \sin \varphi - 4 \cdot \sin^3 \varphi \]
\[ \cos(3\varphi) = 4 \cdot \cos^3 \varphi - 3 \cdot \cos \varphi \]
These identities reveal deeper insights into the periodic and oscillatory nature of trigonometric functions and their applications.