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Chapter 1: Problem 19

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

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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

01

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: $$ e^{-i\psi} = \cos\psi - i\sin\psi $$
02

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: $$ \cos\psi = \frac{e^{i\psi}+e^{-i\psi}}{2} $$
03

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: $$ \sin\psi = \frac{e^{i\psi}-e^{-i\psi}}{2i} $$

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