Last updated October 8, 2026 · Report an error
The sum and difference identities let you compute the sine, cosine, or tangent of a sum or difference of two angles (A+B or A−B) directly from the individual sines, cosines, and tangents of A and B alone, without ever needing to know A+B's ratio from a table or a full unit-circle lookup. They're the foundation many other trig identities are built from — the double-angle identities are simply the special case where A and B are equal, and they show up in physics whenever two waves or signals of different phase are combined (interference and superposition problems routinely reduce to a sum-of-angles expression).
How it works
Enter two angles, A and B, and their shared unit (degrees or radians). The calculator applies sin(A±B) = sinAcosB ± cosAsinB for sine, cos(A±B) = cosAcosB ∓ sinAsinB for cosine (note the sign flips between the two), and tan(A±B) = (tanA±tanB)/(1∓tanAtanB) for tangent. All three follow from the geometry of rotating one angle by another on the unit circle, and all three are computed from the same four underlying values — sinA, cosA, sinB, and cosB — so showing the sum and the difference together costs nothing extra.
- Enter angle A.
- Enter angle B.
- Enter angle unit.
- Click Calculate to see your results.
Examples
sin(30° + 45°)
Computed via the identity: sin30°cos45° + cos30°sin45° = (0.5)(0.7071) + (0.8660)(0.7071) ≈ 0.9659, matching sin(75°) computed directly.
cos(45° − 30°)
Computed via cos45°cos30° + sin45°sin30° = (0.7071)(0.8660) + (0.7071)(0.5) ≈ 0.9659, matching cos(15°) computed directly.
tan(45° + 30°)
Computed via (tan45° + tan30°)/(1 − tan45°tan30°) = (1 + 0.5774)/(1 − 0.5774) ≈ 3.732, matching tan(75°) computed directly.
Who should use it
- Deriving an exact trig value for a non-standard angle (like 75° or 15°) from two known reference angles.
- Physics problems combining two waves or signals with a phase difference.
- Trigonometry coursework on angle-sum identities.
Industry applications
- Physics and signal processing
- Trigonometry and pre-calculus education
Advantages
- Computes sine, cosine, and tangent of both the sum and difference in one pass.
- Avoids needing a direct table lookup for A+B or A−B.
Limitations
- Tangent's identity has an undefined edge case that needs separate handling.
Common mistakes to avoid
- Mixing up the ± and ∓ signs between the sine and cosine versions of the identity — sine keeps the same sign as the sum/difference, cosine flips it.
- Assuming sin(A+B) = sinA + sinB — this is a very common but incorrect simplification; the real identity requires the cross terms with cosine.
- Forgetting to check the denominator in the tangent identity for a possible division by zero.
Best practices
- Compute sinA, cosA, sinB, and cosB first, then plug into whichever identity you need — all three share the same four building blocks.
- Use the special case A = B as a quick way to double-check the double-angle formula if you already have this identity memorized.
- Watch for the undefined case in the tangent identity when A+B or A−B lands on 90° or 270°.
Tips
- Need the special case of A = B instead? Use the Double Angle Formula Calculator.