Calculator Inputs
The overall page stays single-column, while the calculator fields use a responsive 3-column, 2-column, and 1-column layout.
Example Data Table
These sample cases show how common angle pairs create exact compound-angle results.
| Angle A | Operation | Angle B | Result Angle | sin(result) | cos(result) | tan(result) |
|---|---|---|---|---|---|---|
| 45° | + | 30° | 75° | (√6 + √2) / 4 | (√6 - √2) / 4 | 2 + √3 |
| 60° | - | 45° | 15° | (√6 - √2) / 4 | (√6 + √2) / 4 | 2 - √3 |
| 30° | + | 60° | 90° | 1 | 0 | Undefined |
| 120° | - | 60° | 60° | √3 / 2 | 1 / 2 | √3 |
| 225° | + | 45° | 270° | -1 | 0 | Undefined |
Formula Used
Compound-angle identities turn a combined angle into products of separate trigonometric values.
Sine identities
sin(A + B) = sinA·cosB + cosA·sinB
sin(A − B) = sinA·cosB − cosA·sinB
Cosine identities
cos(A + B) = cosA·cosB − sinA·sinB
cos(A − B) = cosA·cosB + sinA·sinB
Tangent identities
tan(A + B) = (tanA + tanB) / (1 − tanA·tanB)
tan(A − B) = (tanA − tanB) / (1 + tanA·tanB)
Normalization and reference angle
Normalize the final angle to a standard interval for cleaner interpretation.
Reference angle helps identify the quadrant behavior and related exact values.
How to Use This Calculator
Follow these steps to calculate and interpret a compound angle quickly.
- Enter Angle A and Angle B using the chosen unit.
- Select whether you want to evaluate A + B or A - B.
- Choose a normalization range so the result is displayed in your preferred interval.
- Set the decimal precision and choose a graph scope.
- Enable exact-value mode if you want special-angle symbolic output when available.
- Click the calculate button to show the result above the form.
- Review the trigonometric values, applied identities, and graph.
- Use the CSV or PDF buttons to export the results.
FAQs
1) What is a compound angle?
Compound angles combine two angles through addition or subtraction, such as A + B or A - B. Trigonometric identities then convert the combined angle into expressions using the separate angles.
2) When do exact values appear?
Exact values appear when the final angle matches supported special angles, including common 15°, 30°, 45°, 60°, and quadrant-related forms. Otherwise the page shows precise decimal approximations.
3) Why does normalization matter?
Normalization rewrites the result into a chosen standard interval. That makes comparison easier, keeps graphs readable, and helps identify equivalent angles like 390° and 30°.
4) Why can tangent be undefined?
Tangent is undefined whenever cosine equals zero, such as 90° and 270°. In those positions the denominator of tan θ = sin θ / cos θ becomes zero.
5) Can I use radians instead of degrees?
Yes. Enter angles in radians, switch the unit selector, and the calculator still evaluates the same addition or subtraction identities. The result also appears in both radians and degrees.
6) What does the graph show?
The chart plots sine, cosine, and tangent across either a full cycle or a local window around the computed compound angle. It helps you compare periodic behavior with the final result.
7) Are the decimal results exact?
Decimals are rounded to your chosen precision for display only. Internally, the calculator uses full floating-point values from the server before formatting the visible results.
8) Can I export the result?
Yes. CSV export saves the numeric output for spreadsheets, while PDF export creates a compact report containing the inputs, formulas, angle conversions, and trigonometric results.