Solving Hyperbolic Equations

This lesson covers techniques for solving equations involving hyperbolic functions: using exponential definitions, using identities to simplify, and recognising quadratic forms.

---

1 Using Exponential Definitions

The most direct method: replace sinh, cosh, tanh with their exponential definitions, then solve.

Worked Example 1

Solve sinh x = 2.

(e^x - e^(-x))/2 = 2 leads to e^(2x) - 4e^x - 1 = 0.

Let u = e^x: u = 2 + sqrt(5) (rejecting the negative root).

x = ln(2 + sqrt(5))

---

2 Worked Example 2

Solve cosh x = 3.

e^(2x) - 6e^x + 1 = 0 gives u = 3 +/- 2 sqrt(2).

Both roots are positive, so: x = +/- ln(3 + 2 sqrt(2))

(cosh is even, so two solutions when k > 1)

---

3 Worked Example 3

Solve tanh x = 3/5.

(e^x - e^(-x))/(e^x + e^(-x)) = 3/5

5e^x - 5e^(-x) = 3e^x + 3e^(-x)

2e^x = 8e^(-x), so e^(2x) = 4.

x = ln 2

---

4 Using Identities

For equations with mixed or squared terms, use identities to reduce to one function.

Worked Example 4