Sum and Difference Identities Calculator

How to use this sum and difference identities calculator

What this calculator does

This calculator evaluates one trigonometric function at two related angles: the sum α + β and the difference α – β. It expands each result with the appropriate sum or difference identity, calculates the numerical value, and compares that value with a direct evaluation of the combined angle. The comparison is a useful verification step for homework, lesson preparation, symbolic manipulation, and engineering calculations. It does not produce a formal proof, choose the best angle decomposition for you, or guarantee an exact radical form for arbitrary angles; its displayed results are high-precision decimal evaluations.

When to use it

Use the tool when you need to evaluate a nonstandard angle from familiar angles, verify an identity expansion, check signs in a sine or cosine formula, or inspect whether tangent, cotangent, secant, or cosecant is undefined at a combined angle. The method is especially helpful for angles such as 15°, 75°, and 105°, which can be written using common unit-circle angles. OpenStax gives a detailed, proof-oriented treatment of the sum and difference formulas for sine, cosine, and tangent.

How to calculate

1. The calculator opens with a complete demonstration: Function is Cosine, Angle unit is Degrees, α is 30, and β is 45. The first results and a validated example XLSX workbook are immediately available.
2. Choose the required item in Function. The available functions are sine, cosine, tangent, cotangent, secant, and cosecant.
3. Select Angle unit. Switching between Degrees and Radians converts the existing α and β values, so the represented angles remain the same.
4. Replace α (alpha) and β (beta) with plain decimal numbers. Results update as you type. Read the sum result first, then the difference result and the verification table.
5. Select Download Excel to export the current controls, component values, formulas, results, and validation checks as a real XLSX workbook. Select Reset to clear the demonstration values. Reset leaves the data fields empty and may disable Excel export until both angles contain a complete valid state again.

Input guide

Function is required and accepts one of six named trigonometric functions. Cosine is the startup choice. Changing it replaces the identity, result labels, pole conditions, and workbook values. A common mistake is to assume all six functions are finite for every angle: tangent and secant are undefined where cosine is zero, while cotangent and cosecant are undefined where sine is zero.

Angle unit is required and accepts Degrees or Radians. Degrees use the degree symbol in results; radians use “rad.” For example, 30° converts to approximately 0.5235987756 rad. Do not manually change only the unit after entering a value with a different meaning; use the selector so the calculator performs the conversion.

α (alpha) is a required finite decimal angle. It accepts digits, one decimal point, and an optional leading sign. Example: 30 or – 0.5. It does not accept commas, degree symbols, π notation, or scientific notation. Increasing α changes both α + β and α – β, so both outputs move.

β (beta) follows the same format and is also required. Example: 45. Increasing β increases the sum angle but decreases the difference angle. Negative β is valid and reverses that relationship. A frequent sign error is forgetting that subtracting a negative β becomes addition.

Output guide

Function and Unit pills restate the selected calculation context. The α + β and α – β pills show the two combined angles. The large primary result is the selected function at α + β; the first smaller result is the same function at α – β. Each is an identity-based numerical estimate, not a recommendation. Zero is a valid result for sine, cosine, tangent, or cotangent when their numerator is zero. “Undefined” indicates a genuine pole rather than a calculation failure.

Sum angle and Difference angle repeat the combined arguments with the selected unit. Largest check error is the larger absolute difference between the expanded identity and direct evaluation. It should be zero at the shown precision or extremely small because of floating-point rounding. In the Identity verification table, Operation distinguishes Sum from Difference, Expanded identity shows the applicable formula, Identity value comes from the expansion, Direct value evaluates the combined angle, and Absolute error compares the two paths.

Worked example

With Cosine selected, α = 30° and β = 45°. The sum angle is 75°. The cosine sum identity is:

Using cos 30° = √3/2, cos 45° = √2/2, sin 30° = 1/2, and sin 45° = √2/2 gives (√6 – √2)/4 ≈ 0.2588190451, which matches the first-open primary result. For the difference, cos(30° – 45°) uses a plus sign between the products and gives approximately 0.9659258263. The exact sign pattern and related identities are also summarized by Wolfram MathWorld's trigonometric addition formulas.