Complex definitions of sine and cosine

TL;DR

The video explains how to derive the complex definitions of sine and cosine using Euler's formula.

Transcript

okay I'm gonna show you guys a complex definition of sine and cosine and first welcome start off with the orders formula e to the I theta is equal to cosine theta plus I sine theta and we are going to plug in Z and negative Z into theta and we'll come with two equations first let me plug in C into all the theta so we'll have e to the I Z and this i... Read More

Key Insights

• 👨‍💼 Euler's formula, e to the i theta equals cosine theta plus i sine theta, is essential in understanding the complex definitions of sine and cosine.
• 🤪 Plugging in both Z and -Z into Euler's formula allows for the derivation of separate equations for cosine Z and sine Z.
• 💤 The complex definition of cosine Z is (e to the i Z + e to the -i Z) / 2, which resembles the hyperbolic cosine function.

Questions & Answers

Q: What is Euler's formula and how is it used to derive the complex definitions of sine and cosine?

Euler's formula is e to the i theta equals cosine theta plus i sine theta. By substituting Z and -Z into the formula and manipulating the equations, the complex definitions of cosine Z and sine Z can be derived.

Q: How does plugging in -Z into the equation help find the expressions for cosine and sine?

Plugging in -Z into the equation cancels out the imaginary part of cosine (-Z) and gives the same expression as cosine Z. Similarly for sine (-Z), the negative sign can be brought to the front due to the properties of the Taylor series, resulting in -i sine Z.

Q: What is the relationship between the complex definition of cosine Z and the hyperbolic cosine function?

The complex definition of cosine Z, (e to the i Z + e to the -i Z) / 2, is similar to the formula for the hyperbolic cosine function, cosh(X), which is (e to the X + e to the -X) / 2. The two forms are closely related.

Q: How can the complex definition of sine Z be derived using Euler's formula?

By manipulating the equations derived from Euler's formula, the complex definition of sine Z can be obtained as (e to the i Z - e to the -i Z) / (2i).

Summary & Key Takeaways

• The video introduces Euler's formula, e to the i theta equals cosine theta plus i sine theta.

• By plugging in Z and -Z into the equation, two separate equations for cosine Z and sine Z can be derived.

• Adding and subtracting these equations, the complex definitions of cosine Z and sine Z can be obtained.