Sin (a - b) Formula - Proof & Examples

What Is the Sin (a - b) Formula?

The sin(a − b) formula, also called the sine difference identity, is:

$$\sin(a - b) = \sin a \cos b - \cos a \sin b$$

In words: the sine of a difference of two angles equals the sine of the first times the cosine of the second, minus the cosine of the first times the sine of the second. The angles $a$ and $b$ can be anything; the formula does not require them to be special.

It is the companion of the sum identity $\sin(a + b) = \sin a \cos b + \cos a \sin b$ — only the middle sign changes. Both belong to the sum and difference identities, the family that also gives the cosine version covered in the cos(a − b) formula. If sine and cosine themselves need a refresher, the trigonometric ratios reference defines both before we combine them here.

One Formula, or a Different One?

A reader chasing $\sin a - \sin b$ — the difference of two separate sine values — wants a different identity entirely:

$$\sin a - \sin b = 2\cos\dfrac{a + b}{2},\sin\dfrac{a - b}{2}$$

That is the sum-to-product form. The sin(a − b) formula on this page is for the sine of one combined angle $(a - b)$. Telling these two apart is the whole game.

How Do You Prove the Sin (a - b) Formula?

There are two clean routes. The geometric one is in the diagram above; the algebraic one is shorter and worth seeing, because it shows the difference formula is just the sum formula with $b$ replaced by $-b$.

Step 1 — Start from the sum identity

$$\sin(a + b) = \sin a \cos b + \cos a \sin b$$

Step 2 — Replace $b$ with $-b$

$$\sin(a - b) = \sin\big(a + (-b)\big) = \sin a \cos(-b) + \cos a \sin(-b)$$

Step 3 — Apply even/odd symmetry

Cosine is even, so $\cos(-b) = \cos b$. Sine is odd, so $\sin(-b) = -\sin b$:

$$\sin(a - b) = \sin a \cos b + \cos a (-\sin b)$$

Step 4 — Simplify

$$\boxed{\sin(a - b) = \sin a \cos b - \cos a \sin b}$$

The minus sign in the formula is a direct fingerprint of sine being an odd function. Sine's odd symmetry is exactly what flips the sign that the sum formula carries as a plus.

Examples of Sin (a - b)

Example 1: Find $\sin 15°$ by writing $15° = 45° - 30°$

Apply the formula with $a = 45°$, $b = 30°$:

$$\sin 15° = \sin 45°\cos 30° - \cos 45°\sin 30°$$

Substitute $\sin 45° = \cos 45° = \dfrac{1}{\sqrt{2}}$, $\cos 30° = \dfrac{\sqrt{3}}{2}$, $\sin 30° = \dfrac{1}{2}$:

$$\sin 15° = \dfrac{1}{\sqrt{2}}\cdot\dfrac{\sqrt{3}}{2} - \dfrac{1}{\sqrt{2}}\cdot\dfrac{1}{2} = \dfrac{\sqrt{3} - 1}{2\sqrt{2}}$$

Rationalising gives $\dfrac{\sqrt{6} - \sqrt{2}}{4}$.

Final answer: $\sin 15° = \dfrac{\sqrt{6} - \sqrt{2}}{4}$.

Example 2: Evaluate $\sin(60° - 30°)$. Wrong path first

The tempting move is to subtract the sines: $\sin 60° - \sin 30° = \dfrac{\sqrt{3}}{2} - \dfrac{1}{2} = \dfrac{\sqrt{3} - 1}{2} \approx 0.37$.

Check it. $60° - 30° = 30°$, and $\sin 30° = \dfrac{1}{2} = 0.5$. The subtraction gave $0.37$, not $0.5$, so subtracting the sines is wrong.

The rescue is the formula:

$$\sin(60° - 30°) = \sin 60°\cos 30° - \cos 60°\sin 30°$$ $$= \dfrac{\sqrt{3}}{2}\cdot\dfrac{\sqrt{3}}{2} - \dfrac{1}{2}\cdot\dfrac{1}{2} = \dfrac{3}{4} - \dfrac{1}{4} = \dfrac{1}{2}$$

Final answer: $\dfrac{1}{2}$, which matches $\sin 30°$ exactly.

Example 3: Simplify $\sin 70°\cos 10° - \cos 70°\sin 10°$

The expression matches the formula with $a = 70°$, $b = 10°$:

$$= \sin(70° - 10°) = \sin 60° = \dfrac{\sqrt{3}}{2}$$

Final answer: $\dfrac{\sqrt{3}}{2}$.

Example 4: Given $\sin a = \dfrac{3}{5}$ (Quadrant I) and $\sin b = \dfrac{5}{13}$ (Quadrant I), find $\sin(a - b)$

First find the cosines. Since both angles are in Quadrant I, cosine is positive:

$$\cos a = \sqrt{1 - \tfrac{9}{25}} = \dfrac{4}{5}, \qquad \cos b = \sqrt{1 - \tfrac{25}{169}} = \dfrac{12}{13}$$

Apply the formula:

$$\sin(a - b) = \dfrac{3}{5}\cdot\dfrac{12}{13} - \dfrac{4}{5}\cdot\dfrac{5}{13} = \dfrac{36}{65} - \dfrac{20}{65} = \dfrac{16}{65}$$

Final answer: $\dfrac{16}{65}$.

Example 5: Prove $\sin(40° + \theta)\cos(10° + \theta) - \cos(40° + \theta)\sin(10° + \theta) = \dfrac{\sqrt{3}}{2}$

The left side fits the formula with $a = 40° + \theta$, $b = 10° + \theta$:

$$= \sin\big((40° + \theta) - (10° + \theta)\big) = \sin 30°$$

Wait — recheck the constants: $(40° + \theta) - (10° + \theta) = 30°$, and $\sin 30° = \dfrac{1}{2}$.

Final answer: the expression simplifies to $\sin 30° = \dfrac{1}{2}$.

Example 6: Show that $\sin(a - b) + \sin(a + b) = 2\sin a\cos b$

Add the two identities, lining up terms:

$$\sin(a - b) = \sin a\cos b - \cos a\sin b$$ $$\sin(a + b) = \sin a\cos b + \cos a\sin b$$

The $\cos a\sin b$ terms cancel:

$$\sin(a - b) + \sin(a + b) = 2\sin a\cos b$$

Final answer: identity proved.

The first-instinct error students reach for across these is subtracting the sines directly — the confusion between $\sin(a - b)$ and $\sin a - \sin b$ is the single most common source of wrong answers in compound-angle work.

Why the Sin (a - b) Formula Matters - "Building unknown angles from known ones"

The formula earns its place because most useful angles are not on any standard table. The trick that unlocks them is rewriting an awkward angle as a sum or difference of friendly ones — $15° = 45° - 30°$, $75° = 45° + 30°$ — and the compound-angle identities turn that rewrite into an exact value.

• Exact values without a calculator - $\sin 15°$, $\sin 75°$, $\sin 105°$ all come straight from these identities, which is why exams test them.

• Wave combination - When two oscillations of different phase add, the sin(a − b) and sin(a + b) identities describe the result — the basis of interference and beat frequencies in physics.

• Rotation - Rotating a point by an angle in the plane uses exactly $\sin(a \pm b)$ and $\cos(a \pm b)$; this is the rotation matrix behind computer graphics.

The deeper point most references skip: the formula is what makes sine predictable under combination. Without it, every new angle would be a fresh measurement instead of a calculation.

Common Mistakes With Sin (a - b)

Mistake 1: Writing sin(a − b) as sin a − sin b

Where it slips in: The moment a difference appears inside sine.

Don't do this: $\sin(a - b) = \sin a - \sin b$.

The correct way: $\sin(a - b) = \sin a\cos b - \cos a\sin b$ — sine and cosine mix; you cannot distribute sine over subtraction.

The point of confusion is treating sine like a quantity that distributes across addition the way a constant multiplier would. Sine is a function, not a factor, and the formula is what a function of a difference actually looks like.

Mistake 2: Flipping the middle sign

Where it slips in: Recalling the formula under exam pressure.

Don't do this: $\sin(a - b) = \sin a\cos b + \cos a\sin b$ (that is the sum formula).

The correct way: The difference formula has a minus in the middle: $\sin a\cos b - \cos a\sin b$. The sign matches the sign in $(a - b)$.

The memorizer who learns "sin cos minus cos sin" by rhythm alone often transplants the plus from the sum identity; the fix is to tie the middle sign to the sign between the angles.

Mistake 3: Getting a cosine sign wrong from the quadrant

Where it slips in: Problems like Example 4, where you compute $\cos a$ from $\sin a$.

Don't do this: Assuming $\cos a$ is positive without checking which quadrant $a$ is in.

The correct way: Use the quadrant to set the sign before substituting — cosine is negative in Quadrants II and III.

Key Takeaways

• The sin(a − b) formula is $\sin a\cos b - \cos a\sin b$.

• $\sin(a - b)$ is not $\sin a - \sin b$ — sine does not distribute over subtraction.

• It is proved by replacing $b$ with $-b$ in the sum identity.

• Use it to find non-standard angles: $\sin 15° = \dfrac{\sqrt{6} - \sqrt{2}}{4}$.

• The middle sign is minus, matching the sign in $(a - b)$.

To go further with the sin(a − b) formula and a teacher, explore Bhanzu's trigonometry tutor sessions, a high school math tutor for compound-angle practice, or live math classes online with peers from 20+ countries.

A Practical Next Step

Practice these to solidify your understanding: find $\sin 75°$ using $45° + 30°$, then simplify $\sin 80°\cos 20° - \cos 80°\sin 20°$. If you get stuck on the middle sign, come back to the four-step proof. Want a live Bhanzu trainer to check your compound-angle work? Book a free demo class.

Read More

• Sin A + Sin B formula — the sum-to-product identity for separate sines.

• 2 sin a cos a formula — a product-to-sum companion identity.

• Trigonometric identities — the broader identity toolkit.

• Cofunction identities — how sine and cosine convert into each other.

• Sin Cos Tan — the three primary ratios these formulas combine.