If sech(u) = 3/5 find the other five hyperbolic functions of u by using identities

TL;DR

Given secant of u = 3/5, find hyperbolic functions using reciprocal relationships.

Transcript

in this problem we're told that the hyperbolic secant of u is equal to three fifths and we're being asked to find the other five hyperbolic functions so let's start by finding the hyperbolic cosine of u so the hyperbolic cosine of u is just the reciprocal of hyperbolic secant so it'll just be five thirds you just flip it okay that's just because um... Read More

Key Insights

• ❓ Reciprocal relationships between hyperbolic functions simplify calculations.
• 🦻 Identity cos^(2)(u) - sinh^(2)(u) = 1 aids in finding hyperbolic sine.
• 🗂️ Hyperbolic tangent is obtained by dividing hyperbolic sinh by hyperbolic cosine.
• ❓ Hyperbolic cotangent is found by reciprocating the hyperbolic tangent.

Questions & Answers

Q: How do we find the hyperbolic cosine of u?

The hyperbolic cosine of u is found by taking the reciprocal of the hyperbolic secant, giving us 5/3.

Q: What identity is used to find the hyperbolic sine of u?

The identity cos^(2)(u) - sinh^(2)(u) = 1 is used to find the hyperbolic sine of u as ±4/3.

Q: How is the hyperbolic tangent of u calculated?

The hyperbolic tangent of u is calculated by dividing hyperbolic sinh by hyperbolic cosine, resulting in ±4/5.

Q: Explain how to find the hyperbolic cotangent of u.

The hyperbolic cotangent of u is simply the reciprocal of the hyperbolic tangent, giving us ±5/4.

Summary & Key Takeaways

• Given secant(u) = 3/5, find hyperbolic cosine as 5/3 using reciprocal relationship.

• Use identity cos^(2)(u) - sinh^(2)(u) = 1 to find sinh(u) as ±4/3.

• Determine hyperbolic tangent as ±4/5 and hyperbolic cotangent as ±5/4.