Calculator Input
Example Data Table
| Angle | Unit | Terms | Mode | Approximate cos(x) |
|---|---|---|---|---|
| 0.5 | Radians | 4 | Maclaurin | 0.877604 |
| 1.2 | Radians | 6 | Maclaurin | 0.362358 |
| 60 | Degrees | 5 | Maclaurin | 0.500000 |
| 45 | Degrees | 4 | Taylor at 30° | 0.707103 |
Formula Used
The calculator supports two cosine series models. The first is the Maclaurin series, centered at zero. The second is the Taylor series, centered at any angle you choose.
For Taylor expansion, the derivatives cycle in this order: cos(a), -sin(a), -cos(a), sin(a), then repeat. Each added term usually improves accuracy, especially when x stays near the expansion center.
Absolute error is |exact value - approximation|. Relative error is absolute error divided by the exact value, shown as a percentage when the exact value is nonzero.
How to Use This Calculator
- Enter the angle value you want to evaluate.
- Select radians or degrees as the input unit.
- Choose the number of series terms to include.
- Select Maclaurin for expansion about zero, or Taylor for another center.
- If Taylor mode is selected, enter the expansion center.
- Choose your preferred decimal precision.
- Press Calculate Cosine Series to show the result above the form.
- Use the CSV or PDF buttons to export the calculated output.
Frequently Asked Questions
1. What does this calculator compute?
It estimates cosine using a finite series expansion. It also shows the exact cosine value, absolute error, relative error, and every partial sum used in the approximation.
2. What is the difference between Maclaurin and Taylor modes?
Maclaurin expands around zero only. Taylor expands around any chosen center. Taylor mode can converge faster when your input angle is close to that selected center.
3. Why do more terms improve accuracy?
Each additional term captures more of the cosine curve. For many inputs, the partial sums move closer to the exact value as the number of included terms increases.
4. Can I enter degrees instead of radians?
Yes. The calculator accepts either unit. Degree inputs are converted internally to radians before the cosine series is evaluated.
5. When should I use Taylor mode?
Use Taylor mode when your angle is near a convenient center, such as 30°, 45°, or π/2. That often produces a better approximation with fewer terms.
6. What range of terms can I use?
You can choose from 1 to 20 terms. Higher values usually improve accuracy, but very large powers and factorials can create tiny numerical rounding differences.
7. What does the term table show?
It lists the index, general term expression, individual term value, and running partial sum. This helps you study convergence and verify each approximation step clearly.
8. What do the export buttons include?
The CSV export includes summary metrics and term values. The PDF export creates a compact report with the key inputs, approximation, exact value, and errors.