Learn How to Find the Hyperbolic Sine and Hyperbolic Tangent Given the Hyperbolic Cosine

TL;DR

Given cosh(x) = 5/4, find sinh(x) and tanh(x) using trigonometric identities.

Transcript

hi everyone in this problem we have that the hyperbolic cosine of x is equal to five over four and we have to find the cinch of x and the tangent of x let's go ahead and go through it so we'll start by using an identity so if you have the hyperbolic cosine squared of x minus the hyperbolic sine squared of x that is equal to one so we can take this ... Read More

Key Insights

• ❓ Trigonometric identities can be used to relate hyperbolic functions like sinh and cosh in problem-solving.
• 🆘 Understanding the reciprocal property helps simplify calculations involving fractions in trigonometry.
• 🤝 The concept of plus or minus solutions arises when dealing with square roots and variable terms in equations.

Questions & Answers

Q: What is given in the problem that needs to be solved?

The problem provides cosh(x) = 5/4 and asks to find sinh(x) and tanh(x) using trigonometric identities.

Q: How is the identity cosh^2(x) - sinh^2(x) = 1 used in the solution?

The identity is used to establish a relationship between cosh(x) and sinh(x) leading to the solution of sinh(x) as plus or minus 3/4.

Q: How is tanh(x) calculated in the problem?

Tanh(x) is calculated by dividing sinh(x) by cosh(x) and simplifying the expression to get plus or minus 3/5 as the possible values.

Q: Why are there plus or minus signs in the solutions for sinh(x) and tanh(x)?

The plus or minus signs account for the two possible solutions due to the nature of trigonometric functions and square roots in the problem.

Summary & Key Takeaways

• Given cosh(x) = 5/4, find sinh(x) and tanh(x) using trigonometric identities.

• Using the identity cosh^2(x) - sinh^2(x) = 1, solve for sinh(x) to be plus or minus 3/4.

• Calculate tanh(x) by dividing sinh(x) by cosh(x) to get plus or minus 3/5 as the possible values.