In one line
A transcendental equation contains non-polynomial functions such as \(e^x\), \(\log x\), \(\sin x\) (e.g. \(x=\cos x\)); we locate a root by finding two values \(a,b\) where \(f(a)\) and \(f(b)\) have opposite signs.
Explained simply
Equations like \(x^2-5x+6=0\) have a formula for their roots. But \(x=\cos x\) or \(xe^x=2\) do not; no algebra can isolate \(x\). Engineers meet such equations all the time, for example in heat transfer, vibrations and circuit design. So we use numerical methods that home in on the root step by step, like a guessing game with "hot" and "cold" hints.
Technically, a transcendental equation is \(f(x)=0\) where \(f\) involves exponential, logarithmic, trigonometric or other non-algebraic functions. High-degree polynomial equations (like \(x^3-2x-5=0\)) are also solved by the same numerical methods. The starting point is the intermediate value property: if \(f\) is continuous on \([a,b]\) and \(f(a)f(b)<0\), there is at least one root between \(a\) and \(b\).
How it works
- Write the equation as \(f(x)=0\).
- Make a small table of \(f(x)\) at simple values like \(0,1,2,3\) (in radians for trig functions).
- Find two consecutive values where the sign changes.
- That interval contains a root. Pick a starting value (often the end where \(|f|\) is smaller).
- Apply Newton–Raphson or Regula Falsi to refine the root.
Key formula
If \(f\) is continuous on \([a,b]\) and \(f(a)\cdot f(b)<0\), then \(f(c)=0\) for some \(c\in(a,b)\).
Quick example
Locate a root of \(f(x)=x-\cos x\) (radians).
| \(x\) | 0 | 0.5 | 1 |
|---|---|---|---|
| \(f(x)\) | \(-1\) | \(0.5-0.8776=-0.3776\) | \(1-0.5403=0.4597\) |
The sign changes between 0.5 and 1, so a root lies in \((0.5,1)\). (The actual root is about 0.7391.)
Common mistakes
- Using degrees instead of radians on the calculator for \(\sin\), \(\cos\), \(\tan\).
- Assuming exactly one root in the interval; a sign change guarantees at least one, possibly more.
- Choosing a huge interval, which makes the iteration slow.
How MU asks it
Not asked as a separate question, but every Newton–Raphson or Regula Falsi question (usually 5/6 marks) begins with it. Show the table of \(f(x)\) values with signs, state the interval, then begin the iteration. Many questions say "correct to three decimal places", so carry at least four decimals in working.