Hyperbolic equations

Objective. Solve equations using identities and exponentials.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

Equations involving hyperbolic functions are solved either by using identities to reduce them to a single hyperbolic function, or by converting to exponential form and solving as an equation in eˣ, often a disguised quadratic. For example, solving cosh x = 2 directly uses the logarithmic inverse form x = arcosh 2 = ln(2+√3), while an equation like 2sinh²x + 3cosh x = 0 first uses cosh²x−sinh²x=1 to rewrite entirely in cosh x before solving the quadratic. Checking the range of each hyperbolic function (cosh x ≥ 1 always) can eliminate invalid solutions.

Worked example

Solve the equation 2cosh²x − sinh x − 2 = 0 for real x, giving your answer in logarithmic form.

1 $Rewrite cosh²x = 1+sinh²x: 2(1+sinh²x) - sinh x - 2 = 0$
2 $Simplify: 2sinh²x - sinh x = 0, so sinh x(2sinh x - 1) = 0$
3 $Either sinh x = 0, giving x = 0, or sinh x = 1/2$
4 $Using sinh⁻¹x = ln(x+\sqrt(x²+1)), sinh x=1/2 gives x = ln(1/2 + \sqrt(5)/2) = ln((1+\sqrt5)/2)$

Answer: x = 0 or x = ln((1+√5)/2)

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students forget that cosh x ≥ 1 for all real x, so an equation requiring cosh x < 1 has no real solution, and keep searching for a nonexistent answer. Always check the solution lies within the function's range before finalising.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard