Solving hyperbolic equations

Method 1: exponentials

For a cosh x + b sinh x = c, replace cosh x and sinh x by their definitions, multiply through by 2e^x and let u = e^x:

(a + b)u² - 2cu + (a - b) = 0.

Solve for u, reject any u ≤ 0 (since e^x > 0), then x = ln u.

Method 2: identities

If the equation mixes cosh² x and sinh x (or sinh² x and cosh x, or sech and tanh), use cosh² x - sinh² x = 1 to write it in one function, then solve the quadratic.

Use the log forms of the inverse functions for the final answers, for example sinh x = 3 gives x = arsinh 3 = ln(3 + sqrt(10)).

How many solutions?

• cosh x = k has two solutions ±arcosh k if k > 1, one solution (x = 0) if k = 1 and none if k < 1.
• sinh x = k always has exactly one solution.
• tanh x = k has one solution only when -1 < k < 1.

Always check each root against these ranges.

Worked example

Solve 5 cosh x - 3 sinh x = 5, giving exact answers.

1. 5(e^x + e^-x)/2 - 3(e^x - e^-x)/2 = 5
2. Multiply by 2e^x: 5e^2x + 5 - 3e^2x + 3 = 10e^x
3. 2u² - 10u + 8 = 0 with u = e^x, so u² - 5u + 4 = 0 and u = 1 or u = 4

Answer: x = 0 or x = ln 4

Check yourself

1. Solve 3 cosh x + sinh x = 3. One solution is x = 0. Find the other, to 3 decimal places.
   Show answer

   -0.693. 2e^2x - 3e^x + 1 = 0, so (2e^x - 1)(e^x - 1) = 0. e^x = 1/2 gives x = ln(1/2) = -ln 2 = -0.693.

2. Find the positive solution of cosh x = 2, to 3 decimal places.
   Show answer

   1.317. x = arcosh 2 = ln(2 + sqrt(3)) = 1.317. (The other solution is its negative.)

3. How many real solutions does cosh x = 1/2 have?
   • One
   • Two
   • Infinitely many
   • None
   Show answer

   None. cosh x ≥ 1 for all real x, so it can never equal 1/2.

4. Solve 2cosh² x - 5 sinh x = 5. Give the larger solution to 3 decimal places.
   Show answer

   1.818. 2(1 + sinh² x) - 5 sinh x = 5, so 2sinh² x - 5 sinh x - 3 = 0 and (2 sinh x + 1)(sinh x - 3) = 0. sinh x = 3 gives x = ln(3 + sqrt(10)) = 1.818; sinh x = -1/2 gives a negative x.

5. Using u = e^x, which quadratic comes from 4 cosh x + 2 sinh x = 5?
   • 3u² - 5u - 1 = 0
   • u² - 5u + 3 = 0
   • 3u² - 5u + 1 = 0
   • 6u² - 5u + 2 = 0
   Show answer

   3u² - 5u + 1 = 0. Multiply by 2e^x: 4(u² + 1) + 2(u² - 1) = 10u, so 6u² - 10u + 2 = 0, which is 3u² - 5u + 1 = 0.

6. Solve sinh x = 4/3, giving x to 3 decimal places.
   Show answer

   1.099. x = ln(4/3 + sqrt(16/9 + 1)) = ln(4/3 + 5/3) = ln 3 = 1.099.

7. For which value of k does tanh x = k have a solution?
   • k = 1
   • k = -1.5
   • k = 0.9
   • k = 2
   Show answer

   k = 0.9. tanh x takes every value strictly between -1 and 1, and no others.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP3

Last updated 8 November 2026. Spotted a mistake? Tell us.