Inverse Hyperbolic Functions
Logarithmic expressions for arsinh and arcosh.
Key idea
Logarithmic expressions for arsinh and arcosh.
In depth
These are logarithms in disguise: arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² − 1)) for x ≥ 1, artanh x = ½ ln((1 + x)/(1 − x)) for |x| < 1.
Worked example
arsinh 1 = ln(1 + √2) ≈ 0.8814.
Formula
Real-world uses
- Solving differential equations for sensors and engineering models of growth
Common mistake
Ignoring the domain restrictions of arcosh and artanh.
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 Inverse Hyperbolic Functions?
Logarithmic expressions for arsinh and arcosh. It is a topic at the expert level in Trigonometry.
What is the formula for Inverse Hyperbolic Functions?
The key formula is arsinh x = ln(x+√(x²+1)).
Where is Inverse Hyperbolic Functions used in real life?
Real-world examples: Solving differential equations for sensors and engineering models of growth.
What should I learn before Inverse Hyperbolic Functions?
Make sure you are comfortable with Hyperbolic Identities, the previous topic in Trigonometry.
What is a common mistake with Inverse Hyperbolic Functions?
Ignoring the domain restrictions of arcosh and artanh.
How can I practise Inverse 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 Inverse Hyperbolic Functions?
Move on to Fourier Series Introduction 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).