Locate and bisect a circular-equation root
The following circular equation is to be solved numerically
\[\cos x=0.25x + 0.59\]
Write the algebraic function in the equation as \(P(x)=0.25x + 0.59\)
Let \(F(x)\) be the circular side minus the algebraic side, so \(F(x)=\cos x-\left(0.25x + 0.59\right)\)
The sign-changing interval under consideration is \(\left[\frac{2}{5},\frac{9}{10}\right]\). The table also gives one point just outside each endpoint
| \(x\) | \(F(x)\) |
|---|---|
| \(\frac{11}{40}\) | \(\approx 0.304\) |
| \(\frac{2}{5}\) | \(\approx 0.231\) |
| \(\frac{9}{10}\) | \(\approx -0.193\) |
| \(\frac{41}{40}\) | \(\approx -0.327\) |
Use the endpoint signs in the table to identify what can be concluded about a root
Use the evidence to choose the function pair, root count and starting interval