How this instrument works
The sum identities say something specific: the sine and cosine of a combined angle can be rebuilt entirely from the separate sines and cosines of its two parts, without ever measuring A+B itself. sin(A+B) = sinA·cosB + cosA·sinB and cos(A+B) = cosA·cosB − sinA·sinB. The cleanest way to see why the second one carries a minus sign is to write both angles as complex exponentials: e^{iA}·e^{iB} = e^{i(A+B)}. Multiply out the left side, sort the real part from the imaginary part, and cosA·cosB − sinA·sinB falls out as the real component while sinA·cosB + cosA·sinB falls out as the imaginary one — the minus sign is not a rule to memorise, it is where the i·i = −1 term lands.
The difference versions live inside this same calculator for free. Sine is an odd function and cosine is even, so replacing B with −B turns sin(A+B) into sin(A−B) = sinA·cosB − cosA·sinB and cos(A+B) into cos(A−B) = cosA·cosB + sinA·sinB — the signs simply swap. Type a negative value into Angle B and the two output fields become the difference identities without any change to the underlying formula; nothing about the calculator needs to know which case it is in.
Two boundary checks confirm the formula is honest rather than approximate. Set Angle A to zero and sin(0+B) returns sinB exactly, cos(0+B) returns cosB exactly — adding nothing changes nothing. Set the two angles equal instead, A = B, and the general sum formula collapses to the double-angle results sin2A = 2sinA·cosA and cos2A = cos²A − sin²A: those familiar shortcuts are not a separate rule, just this identity with one input repeated.
- Type the first angle into Angle A — the unit control beneath the field defaults to degrees but also accepts radians or turns.
- Type the second angle into Angle B, in whichever unit is convenient; the sheet converts both to a common footing before it multiplies anything.
- Read sin(A + B) — it is assembled purely from sinA, cosB, cosA, and sinB, never by measuring A+B as one angle.
- Read cos(A + B) alongside it; both fields recompute together the moment either angle changes.
- For the difference identities, re-enter Angle B as a negative number: sin(A + B) and cos(A + B) then equal sin(A − B) and cos(A − B) exactly.
Worked example — 30° combined with 45°
Set Angle A to 30° and Angle B to 45°. Internally the sheet works in radians (0.5235987755982988 and 0.7853981633974483), but the identity only ever needs the four separate values sin30° = 0.5, cos30° ≈ 0.8660254, sin45° ≈ 0.7071068, and cos45° ≈ 0.7071068 — the same four numbers a printed trig table would give.
sin(A + B) = (0.5 × 0.7071068) + (0.8660254 × 0.7071068) = 0.9659258262890683, which is sin75° carried to sixteen figures — the actual sine of 30°+45°. cos(A + B) = (0.8660254 × 0.7071068) − (0.5 × 0.7071068) = 0.25881904510252085, matching cos75° just as precisely. Neither figure passed through 75° as an intermediate step; both came from four multiplications and one addition or subtraction.
Questions
How is the sine addition formula actually derived?
The cleanest route uses complex exponentials: e^{iA}·e^{iB} = e^{i(A+B)}. Expanding the left side and separating real and imaginary parts hands you cos(A+B) = cosA·cosB − sinA·sinB as the real part and sin(A+B) = sinA·cosB + cosA·sinB as the imaginary part, in one stroke rather than two separate geometric arguments.
How do I get the difference identities out of this calculator?
Enter Angle B as a negative number. Because cosine is even and sine is odd, cos(A+(−B)) = cosA·cosB + sinA·sinB and sin(A+(−B)) = sinA·cosB − cosA·sinB — exactly the difference forms — appear in the same two output fields with no other change to the formula.
How do these connect to the double-angle formulas?
Set Angle A and Angle B to the same value. The sum formulas then collapse to sin2A = 2sinA·cosA and cos2A = cos²A − sin²A, so the double-angle identities used elsewhere are not separate rules — they are this same sum formula with one angle repeated instead of two distinct ones.
What's the most common mistake people make with sin(A+B)?
Assuming it equals sinA + sinB, treating sine as if it were linear. It is not: sin(30°+30°) = sin60° ≈ 0.866, while sin30° + sin30° = 1. The cross terms cosA·sinB and sinA·cosB in the real formula are exactly what that shortcut throws away.
Do the angles have to be entered in degrees?
No — the unit selector under each angle field defaults to degrees but also accepts radians or turns, and the sheet converts before evaluating either identity. The underlying formula itself is always computed in radians, the natural unit for the sine and cosine functions.
Why does cos(A+B) subtract a term instead of adding it like the sine version does?
It traces back to multiplying e^{iA} by e^{iB}: the imaginary-times-imaginary term sinA·sinB picks up a factor of i² = −1 when it lands in the real part, while the sine identity's cross terms both land in the imaginary part with no such sign flip.