Sum and Difference Formulas
The sum and difference formulas let you find exact values for angles that don’t appear on the unit circle, simplify trigonometric expressions, and prove identities. They also serve as the foundation for the double angle and half angle formulas that appear throughout precalculus and calculus. The formulas themselves aren’t difficult — but the cosine versions have a sign pattern that catches nearly every student at least once.
The Six Formulas at a Glance
sin(A+B) = sin A cos B + cos A sin B
sin(A-B) = sin A cos B – cos A sin B
cos(A+B) = cos A cos B – sin A sin B
cos(A-B) = cos A cos B + sin A sin B
tan(A+B) = (tan A + tan B) / (1 – tan A tan B)
tan(A-B) = (tan A – tan B) / (1 + tan A tan B)
Table of Contents
1) What the Sum and Difference Formulas Are
The sum and difference formulas expand trigonometric functions of sums or differences of two angles into expressions involving the individual angles. Instead of evaluating sin(75°) directly — which the unit circle doesn’t give you — you write it as sin(45° + 30°) and apply the formula to get an exact answer using values you already know.
There are six formulas in total: sum and difference versions for sine, cosine, and tangent. The sine and cosine formulas are the most commonly tested. The tangent formulas appear more often in identity proofs than in direct calculation problems.
Why these formulas exist
The unit circle gives exact values for multiples of 30° and 45° (and their radian equivalents). It does not give you 15°, 75°, 105°, or most other angles. The sum and difference formulas let you express those angles as combinations of angles the unit circle does provide, then calculate an exact result rather than a decimal approximation. That exact form — involving radicals and fractions — is what most precalculus and calculus courses require.
These formulas are not approximations. They are algebraic identities that hold exactly for every pair of angles A and B.
2) How to Use Them: Worked Examples
Finding an exact value for a non-standard angle
The most common use is finding sin, cos, or tan of an angle that isn’t on the unit circle by splitting it into two angles that are.
Example: Find the exact value of sin(75°).
75° = 45° + 30°. Both of those are on the unit circle.
= sin 45° cos 30° + cos 45° sin 30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
= (√6 + √2) / 4
Example: Find the exact value of cos(15°).
15° = 45° − 30°. Use the cosine difference formula — and remember the sign flips.
= cos 45° cos 30° + sin 45° sin 30° (the sign FLIPS to +)
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
= (√6 + √2) / 4
Note that sin(75°) = cos(15°) — which makes sense because 75° and 15° are complementary angles (they add to 90°).
Simplifying an expression
Sometimes the formula runs in reverse: you’re given an expanded expression and asked to identify which sum or difference formula it came from, then write the compact form.
✓ Correct: The pattern cos A cos B + sin A sin B matches cos(A minus B) because the cosine difference formula flips the sign. So cos 3x cos 2x + sin 3x sin 2x = cos(3x – 2x) = cos(x).
Verifying an identity
Identity verification problems ask you to show that two expressions are equal by transforming one side into the other. Sum and difference formulas give you the expansion needed to bridge unfamiliar forms.
Left side: sin(x + π)
= sin x cos π + cos x sin π
= sin x (-1) + cos x (0)
= -sin x ✓
This works because cos(π) = −1 and sin(π) = 0 — values that come directly from the unit circle. See the Unit Circle guide if those values aren’t immediately familiar.
3) The Cosine Sign Flip: Where Students Go Wrong
The sine sum and difference formulas behave intuitively: the sign between the two terms in the expansion matches the sign in the original angle. sin(A+B) has a plus between its terms. sin(A−B) has a minus. There is no trick to learn.
The cosine formulas are the opposite. The sign between the two terms in the expansion always flips from the sign in the original angle. cos(A+B) has a minus between its terms despite the plus in (A+B). cos(A−B) has a plus between its terms despite the minus in (A−B). This flip is non-negotiable and applies every time.
Why the cosine formula flips
The flip isn’t arbitrary — it comes from how cosine behaves as an even function and how it relates to the unit circle. The algebraic derivation using the distance formula and the Pythagorean identity produces the minus sign in cos(A+B) naturally. But for exam purposes, the explanation matters less than the habit: whenever you write a cosine sum or difference formula, the sign between the two terms is the opposite of what’s in the angle.
✓ Correct: cos(A+B) = cos A cos B – sin A sin B. The sign flips. Always.
The pattern across all four cosine and sine formulas
One way to keep the patterns straight is to notice that cos(A+B) and cos(A−B) are written with a minus and plus respectively — they mirror each other. The same mirroring happens in sine: sin(A+B) uses plus, sin(A−B) uses minus. The difference is that for sine the mirror matches the input, and for cosine the mirror is flipped from the input.
4) Connection to Double Angle and Other Identities
The sum formulas are the parent formulas for most of the other identities in a precalculus or trig course. Once you have the sum formulas memorized, you can derive the others rather than memorizing them separately.
Double angle formulas
Set A = B in the sum formulas and you get the double angle formulas immediately.
= sin A cos A + cos A sin A
= 2 sin A cos A
cos(2A) = cos(A + A)
= cos A cos A – sin A sin A
= cos²A – sin²A
The cos(2A) form can be further rewritten using the Pythagorean identity into two alternative forms: 2cos²A − 1 and 1 − 2sin²A. Those come from substituting sin²A = 1 − cos²A and cos²A = 1 − sin²A respectively. See the Pythagorean Identities guide for those substitutions.
Half angle formulas
The half angle formulas are derived by solving the double angle cosine formula for sin²A and cos²A and then taking square roots. They appear in calculus integration problems more often than in precalculus, but some trig courses introduce them alongside the double angle formulas. The derivation goes: cos(2A) = 1 − 2sin²A, solve for sin²A to get sin²A = (1 − cos 2A)/2, take the square root to get sin(A/2) = ±√((1 − cos A)/2).
Identity verification involving multiple formulas
Harder identity problems combine sum/difference formulas with Pythagorean identities and reciprocal identities in a single proof. The approach is always the same: expand the sum or difference formula first, then apply Pythagorean substitutions, then simplify. Working only on one side at a time prevents algebraic errors and is required by most instructors.
Where this shows up on exams
In precalculus, sum and difference formulas appear in three question types: exact value calculations (find the exact value of sin 75°), expression simplification (write as a single trig function), and identity verification (show that the two sides are equal). On timed tests, the cosine sign flip is the highest-frequency error. The sign pattern SVG above is designed to be memorized before any exam where these appear.
5) How FMMC Can Help
Sum and difference formulas appear in precalculus, trigonometry, and college algebra courses across every major online platform — MyMathLab, WebAssign, ALEKS, Hawkes Learning, and others. If you’re working through a trig unit and the identity proofs or exact value problems are holding up your grade, FMMC’s math experts handle every assignment, quiz, and exam with an A/B guarantee.
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Other guides in this series
This guide is part of FMMC’s trigonometry foundation series. The other guides cover:
Unit Circle Explained •
SOH-CAH-TOA •
Radian and Degree Conversion •
Trig Functions Explained •
Inverse Trig Functions •
Pythagorean Identities •
Solving Trig Equations
FAQ: Sum and Difference Formulas
What are the sum and difference formulas used for?
Sum and difference formulas are used to find exact trigonometric values for angles that don’t appear directly on the unit circle (such as 15°, 75°, or 105°), to simplify trigonometric expressions, and to verify identities. They are also the starting point for deriving the double angle and half angle formulas.
Why does cos(A+B) have a minus sign instead of a plus?
The cosine formulas always flip the sign from what appears in the angle. cos(A+B) = cos A cos B − sin A sin B uses a minus even though the angle has a plus. cos(A−B) = cos A cos B + sin A sin B uses a plus even though the angle has a minus. This flip comes from the algebraic derivation of the formula and is not an exception — it is the rule. “Sine stays, cosine changes” is the memory device.
How do you find the exact value of sin(75°)?
Write 75° = 45° + 30°. Then apply the sine sum formula: sin(75°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4. The same approach works for any angle that can be expressed as a sum or difference of 30°, 45°, 60°, 90°, or their multiples.
Are the sum and difference formulas the same as the double angle formulas?
No, but the double angle formulas are derived from the sum formulas. Setting A = B in sin(A+B) gives sin(2A) = 2 sin A cos A. Setting A = B in cos(A+B) gives cos(2A) = cos²A − sin²A. The sum and difference formulas are the parent identities; the double angle formulas are a special case.
How do you remember which formula is which?
For sine: the sign between the two terms in the expansion always matches the sign in the original angle. sin(A+B) uses plus; sin(A−B) uses minus. For cosine: the sign always flips. cos(A+B) uses minus; cos(A−B) uses plus. Memory device: “sine stays, cosine changes.” For tangent: the signs in the numerator and denominator are always opposite — if the numerator uses plus, the denominator uses minus, and vice versa.
Do these formulas work in radians?
Yes. The sum and difference formulas are algebraic identities that hold for any angles A and B regardless of whether those angles are measured in degrees or radians. The unit circle values you substitute in are the same either way — just expressed in radian form (π/6 instead of 30°, π/4 instead of 45°, etc.).
Can FMMC help with trig identity proofs and exam preparation?
Yes. FMMC handles trig homework, quizzes, and proctored exams across MyMathLab, WebAssign, ALEKS, and Hawkes Learning. Identity proofs, exact value problems, and sum/difference formula applications are all within scope. Get a free quote or email info@finishmymathclass.com.
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