Hyperbolic Functions
Analogues of trig functions defined with exponentials.
Key idea
Analogues of trig functions defined with exponentials.
In depth
sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh/cosh. They parametrise the hyperbola x² − y² = 1 as trig functions parametrise the circle. cosh describes a hanging chain (catenary).
Worked example
sinh 0 = 0, cosh 0 = 1, and cosh x ≥ 1.
Formula
Real-world uses
- Describing the shape of a hanging cable or chain (a catenary) in bridge and building design
Common mistake
Assuming they are periodic like sin and cos.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Hyperbolic Functions?
Analogues of trig functions defined with exponentials. It is a topic at the expert level in Trigonometry.
What is the formula for Hyperbolic Functions?
The key formula is sinh x = (eˣ − e⁻ˣ)/2.
Where is Hyperbolic Functions used in real life?
Real-world examples: Describing the shape of a hanging cable or chain (a catenary) in bridge and building design.
What should I learn before Hyperbolic Functions?
Make sure you are comfortable with Euler’s Identity, the previous topic in Trigonometry.
What is a common mistake with Hyperbolic Functions?
Assuming they are periodic like sin and cos.
How can I practise Hyperbolic Functions?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Hyperbolic Functions?
Move on to Hyperbolic Identities next.
Related expert topics in Trigonometry
- Half Angle Formulas – Trig of θ/2.
- Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas – Convert products into sums.
- Sum-to-Product Formulas – Convert sums into products.
- Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).