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Sum and Difference Trig Calculator

Enter two angles and pick sum or difference to evaluate the trig identities.

How it is calculated

Both angles A and B are entered in degrees and converted to radians internally. For the sum option the tool uses sin(A+B) = sinA·cosB + cosA·sinB and cos(A+B) = cosA·cosB − sinA·sinB; for the difference option it uses sin(A−B) = sinA·cosB − cosA·sinB and cos(A−B) = cosA·cosB + sinA·sinB. The tangent is then taken as tan = sin ÷ cos of the combined angle. When the cosine is practically zero (the combined angle is 90°, 270° and so on), the tangent does not exist and the tool shows a dash instead of a number. All three values are rounded to six decimals, so exact values such as 0 or 1 appear as 0.000000 and 1.000000. Any angle is accepted, including negative angles and angles above 360°.

Worked example

Find the values for 45° + 30°, which is 75°. Step 1: the individual values are sin 45° = 0.707107, cos 45° = 0.707107, sin 30° = 0.5, cos 30° = 0.866025. Step 2: sin(A+B) = 0.707107 × 0.866025 + 0.707107 × 0.5, which the tool rounds to 0.965926. Step 3: cos(A+B) = 0.707107 × 0.866025 − 0.707107 × 0.5 gives 0.258819. Step 4: dividing the sine by the cosine gives tan = 3.732051.

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Common mistakes

  • Entering radians. The angle fields are in degrees, so typing 0.785 for π/4 is treated as less than one degree, not 45°.
  • Leaving the option on sum when a difference is wanted. For 45° and 30°, the sum gives sin = 0.965926 but the difference gives sin = 0.258819.
  • Assuming sin(A+B) equals sinA + sinB. The identity has a cross term, and for 45° and 30° the plain sum 0.707107 + 0.5 = 1.207107 is even above 1, which no sine can be.
  • Reading the dash for tan as an error. When A+B or A−B is 90° or 270° the tangent is undefined, which the tool shows with a dash (for 60° + 30°, sin is 1.000000 and cos is 0.000000).
  • Mixing up the signs in the cosine formula. The cosine of a sum subtracts the product of the sines, and the cosine of a difference adds it.

Sources

About this calculator

The sum and difference identities express the sine, cosine and tangent of A±B in terms of the sines, cosines and tangents of A and B individually: sin(A±B) = sinA·cosB ± cosA·sinB, cos(A±B) = cosA·cosB ∓ sinA·sinB, and tan(A±B) = (tanA ± tanB) / (1 ∓ tanA·tanB). This calculator evaluates all three directly from two input angles.

Trigonometry and precalculus students use these identities to simplify expressions and prove other trig identities without a calculator, and engineers working with waveforms and signal processing use them when combining or shifting phases of two sinusoidal signals.

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Comments (1)

  • Editorial team

    These identities are handy for angles that are not on the standard list but split into ones that are. Take 75 degrees as 45 plus 30: the tool gives sin = 0.965926 and cos = 0.258819, with no table lookup needed. Because every value is rounded to six decimals, compare answers to that precision rather than expecting the exact surds. Which non-standard angles do you usually need?

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