Enter two angles (α and β) and choose a trigonometric function — the Sum and Difference Identities Calculator applies the corresponding sum and difference formulas to compute sin(α±β), cos(α±β), or tan(α±β). You get back the exact decimal values for both the sum and difference expressions, plus a breakdown of each formula's components.
f(α + β)
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f(α − β)
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f(α)
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f(β)
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Sum Formula Used
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Difference Formula Used
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Sum and difference identities are formulas that express the sine, cosine, or tangent of the sum or difference of two angles in terms of the trig functions of the individual angles. For example, sin(α + β) = sin(α)cos(β) + cos(α)sin(β). These identities are especially useful when the combined angle does not have a known simple value but the individual angles do.
You can express 15° as 45° − 30°, both of which are standard angles with known cosine and sine values. Applying the cosine difference formula: cos(45° − 30°) = cos(45°)cos(30°) + sin(45°)sin(30°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2) / 4 ≈ 0.9659.
The cosine difference formula states that the cosine of the difference of two angles equals the product of their cosines plus the product of their sines. In symbols: cos(α − β) = cos(α)cos(β) + sin(α)sin(β). Notice the plus sign — this is the key distinction from the sine difference formula.
The sine addition formula is sin(α + β) = sin(α)cos(β) + cos(α)sin(β). Similarly, the sine difference formula is sin(α − β) = sin(α)cos(β) − cos(α)sin(β). These allow you to find the sine of any compound angle using simpler, known values.
The tangent sum formula is tan(α + β) = (tan(α) + tan(β)) / (1 − tan(α)tan(β)), and the tangent difference formula is tan(α − β) = (tan(α) − tan(β)) / (1 + tan(α)tan(β)). These formulas are undefined when the denominator equals zero.
These identities appear in physics (wave interference, signal processing), engineering (circuit analysis, mechanical vibrations), computer graphics (rotation transformations), and navigation. They are fundamental tools whenever angles need to be combined or decomposed.
While memorizing the core sine and cosine addition formulas helps, many other identities (double angle, half angle, product-to-sum) can be derived from them. In practice, calculators and reference tables are commonly used, but understanding the underlying formulas aids problem-solving and verification.
Yes. The sum and difference identities work for any angle values, including negative angles and angles greater than 90°. The calculator accepts values from −360° to 360° (or the equivalent in radians) for both α and β.