Problem 1 Based on Hyperbolic Functions | Summary and Q&A

TL;DR

Learn about circular and hyperbolic functions, their formulas, and how to derive the relation between them.

Key Insights

• #️⃣ Circular functions are derived from Euler's formula, which connects complex numbers to trigonometric functions.
• ❓ Hyperbolic functions are defined using exponential functions and have different properties than circular functions.
• 👻 The relation between circular and hyperbolic functions allows for conversions between the two.
• 😘 Hyperbolic functions can be written in terms of cos and sin functions, where z is a complex number.

Transcript

Read and summarize the transcript of this video on Glasp Reader (beta).

Questions & Answers

Q: What is the formula for circular functions?

The formulas for circular functions are cos theta = (e^(iθ) + e^(-iθ)) / 2 and sin theta = (e^(-iθ) - e^(iθ)) / (2i).

Q: How are hyperbolic functions defined?

Hyperbolic functions are defined as cosh z = (e^z + e^(-z)) / 2, sinh z = (e^z - e^(-z)) / 2, and tanh z = sinh z / cosh z.

Q: What is the relation between sine hyperbolic and cosine hyperbolic?

The relation between sine hyperbolic (sinh z) and cosine hyperbolic (cosh z) is given by tanh z = sinh z / cosh z.

Q: How can circular functions be written in terms of complex numbers?

Circular functions can be written in terms of complex numbers by replacing theta with z, where z = x + iy. For example, cos theta becomes (e^(iz) + e^(-iz)) / 2.

Summary & Key Takeaways

• Circular functions are derived from Euler's formula, while hyperbolic functions are expressed in terms of cos and sin functions.

• Circular functions: cos theta = (e^(iθ) + e^(-iθ)) / 2, sin theta = (e^(-iθ) - e^(iθ)) / (2i)

• Hyperbolic functions: cosh z = (e^z + e^(-z)) / 2, sinh z = (e^z - e^(-z)) / 2, tanh z = sinh z / cosh z