Sine sum and difference is one of 5 angle formulas formulas in the trigonometry section of this library, and it is used at high school · ap level.
Why sine sum and difference works
Sine is not a function you can distribute over addition, because turning by A and then by B moves a point around a curve rather than along a line. Rotating the unit-circle point twice mixes both coordinates of the first position, and the two products sin A cos B and cos A sin B are exactly the two pieces that mixing produces.
What each symbol means
$A,B$ are angles in the same unit.
Sine sum and difference: when it holds
Keep the same sign as the angle operation.
When it stops applying
The formula assumes A and B carry the same unit. Writing sin(45° + π/6) and then evaluating everything in degree mode treats π/6 as about 0.524 of a degree, so you get the sine of 45.52 degrees instead of the sine of 75 degrees, an error of 0.253.
Sine sum and difference: a worked example
$\sin75^\circ=\sin(45^\circ+30^\circ)=\frac{\sqrt6+\sqrt2}{4}$.
The mistake to avoid
What people do: Students write sin(A + B) = sin A + sin B and move straight on.
Why it goes wrong: Test it with A = B = 30 degrees. The short version gives 0.5 + 0.5 = 1, but sin 60 degrees is 0.866, so the shortcut is off by more than 15 percent.
Do this instead: Expand with the full four-term pattern: sin 30 cos 30 + cos 30 sin 30 = 2(0.5)(0.866), which is 0.866. Every sine of a combined angle needs both products, never a plain sum.
Sine sum and difference: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $A,B$ are angles in the same unit.
- Check the conditions before substituting. Keep the same sign as the angle operation.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Angle formulas
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where sine sum and difference comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Trigonometric Identities — the lesson behind this formula: simplify and prove relationships between trig functions.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about sine sum and difference
How do I know which two angles to split 75 degrees into?
Pick a pair from the angles you already know exactly: 30, 45, 60, and 90 degrees. For 75 use 45 + 30, which gives (√6 + √2)/4, about 0.966. For 15 degrees use 45 − 30 instead.
Does the sign in the answer really follow the sign in the angle?
Yes, sine is the friendly one here. A plus between the angles stays a plus between the two products, and a minus stays a minus, which is one less rule than cosine asks you to remember.
Why bother, when a calculator gives sin 75 instantly?
Because the identity gives an exact value rather than a rounded decimal, and because in algebra the angle is often a letter. You cannot type sin(x + 30) into a calculator and simplify it, but you can expand it.
Can I use it with three angles added together?
Yes, by grouping. Treat A + B as one angle and expand sin((A + B) + C), then expand the leftover sin(A + B) and cos(A + B) pieces. It works, but the result runs to eight terms, so group carefully.