Question

Use Euler's Formula to establish the identities \(\cos \psi=\frac{e^{i \psi \nu}+e^{-i \psi}}{2} \quad\) and \(\quad \sin \psi=\frac{e^{i \psi}-e^{-i \psi}}{2 i}\)

Short answer

Question: Use Euler's formula to establish the following trigonometric identities: 1. $\cos\psi = \frac{e^{i\psi}+e^{-i\psi}}{2}$ 2. $\sin\psi = \frac{e^{i\psi}-e^{-i\psi}}{2i}$ Answer: Start by writing down Euler's formula, $e^{i\psi} = \cos\psi + i\sin\psi$, and its complex conjugate, $e^{-i\psi} = \cos\psi - i\sin\psi$. To establish the identity for cosine, add both equations and simplify, resulting in $\cos\psi = \frac{e^{i\psi}+e^{-i\psi}}{2}$. To establish the identity for sine, subtract the complex conjugate equation from Euler's formula and simplify, resulting in $\sin\psi = \frac{e^{i\psi}-e^{-i\psi}}{2i}$.

Step-by-step solution

Write down Euler's formula and its conjugate

First, let's write down Euler's formula, which is: $$ e^{i\psi} = \cos\psi + i\sin\psi $$ Now take the complex conjugate of the equation, which reverses the sign of the imaginary part: $$ e^{-i\psi} = \cos\psi - i\sin\psi $$

Establish the identity for cosine

We will add both equations from Step 1 to establish the identity for cosine: $$ e^{i\psi}+e^{-i\psi} = (\cos\psi + i\sin\psi) + (\cos\psi - i\sin\psi) $$ Simplifying the equation, we get: $$ e^{i\psi}+e^{-i\psi} = 2\cos\psi $$ Now, divide by 2 to obtain the desired identity for cosine: $$ \cos\psi = \frac{e^{i\psi}+e^{-i\psi}}{2} $$

Establish the identity for sine

Similarly, we will subtract the complex conjugate equation from Euler's formula to establish the identity for sine: $$ e^{i\psi}-e^{-i\psi} = (\cos\psi + i\sin\psi) - (\cos\psi - i\sin\psi) $$ Simplifying the equation, we get: $$ e^{i\psi}-e^{-i\psi} = 2i\sin\psi $$ Now, divide by 2i to obtain the desired identity for sine: $$ \sin\psi = \frac{e^{i\psi}-e^{-i\psi}}{2i} $$

Key concepts

Complex Numbers

Complex numbers are a type of number that includes both a real and an imaginary part. A complex number is usually written in the form:

• \( a + bi \)

where \( a \) is the real part and \( b \) is the imaginary part. The imaginary unit \( i \) is defined by the property \( i^2 = -1 \).
Complex numbers are crucial for understanding Euler's Formula, which connects these numbers to trigonometry. A key point about complex numbers is how they can be visualized on the complex plane. In this plane, the horizontal axis represents the real part and the vertical axis represents the imaginary part. Together, these axes enable us to represent complex numbers as points or vectors.
When dealing with complex numbers, we often need to look at their conjugates. The conjugate of \( a + bi \) is \( a - bi \). Using conjugates can simplify many mathematical expressions, just as we used them to establish trigonometric identities using Euler's Formula.

Trigonometric Identities

Trigonometric identities are equations involving the trigonometric functions that are true for every value of the variable involved. These functions include sine, cosine, and tangent. These identities help in simplifying expressions and solving trigonometric equations.
One of the fundamental identities is the Pythagorean identity:

• \( \sin^2\theta + \cos^2\theta = 1 \)

which holds for all angles \( \theta \). Euler's Formula is directly related to these identities because it provides an elegantly simple method to represent sine and cosine functions using complex numbers.
Euler's Formula states:

• \( e^{i\psi} = \cos\psi + i\sin\psi \)

By manipulating this formula and its conjugate, we derive the identities for cosine and sine in terms of exponential functions, effectively bridging the concepts of complex numbers and trigonometry. This is why Euler's Formula is such a powerful tool in mathematical analyses.

Mathematical Derivations

Mathematical derivations are processes that use algebraic manipulations to derive new formulas or identities from known ones. Deriving the trigonometric identities using Euler's Formula is an excellent example of this process.
Euler's Formula, which expresses \( e^{i\psi} \) as \( \cos\psi + i\sin\psi \), allows us to depict complex numbers in terms of trigonometric functions. By strategically adding or subtracting its conjugate \( e^{-i\psi} = \cos\psi - i\sin\psi \), we can isolate terms to express cosine and sine respectively.

• For cosine, adding \( e^{i\psi} \) and \( e^{-i\psi} \) gives \( 2\cos\psi \), which leads to \( \cos\psi = \frac{e^{i\psi} + e^{-i\psi}}{2} \).
• For sine, subtracting \( e^{-i\psi} \) from \( e^{i\psi} \) gives \( 2i\sin\psi \), leading to \( \sin\psi = \frac{e^{i\psi} - e^{-i\psi}}{2i} \).

These derivations not only rely on algebraic manipulation but also utilize the properties of complex conjugates and exponential functions, highlighting the beauty and interconnection of mathematical concepts. Understanding these derivations deepens our comprehension of both trigonometry and complex analysis.