TRIGONOMETRYTrigonometryMathematics Calculator
A±B

Sum and Difference Identities

sin(A±B)=sinAcosB±cosAsinB, cos(A±B)=cosAcosB∓sinAsinB, tan(A±B)=(tanA±tanB)/(1∓tanAtanB). The foundation for double angle and product-to-sum.

Concept Fundamentals
sinAcosB+cosAsinB
sin(A+B)
cosAcosB-sinAsinB
cos(A+B)
(tanA+tanB)/(1-tanAtanB)
tan(A+B)
Set A=B
Double angle

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Double angle: set A=B to get sin(2A)=2sinAcosA, cos(2A)=cos²A-sin²A. sin(A-B)=sinAcosB-cosAsinB — the minus sign applies to the cosAsinB term. tan(A+B) undefined when tanAtanB=1 — i.e., when A+B=90°+n·180°.

Key quantities
sinAcosB+cosAsinB
sin(A+B)
Key relation
cosAcosB-sinAsinB
cos(A+B)
Key relation
(tanA+tanB)/(1-tanAtanB)
tan(A+B)
Key relation
Set A=B
Double angle
Key relation

Ready to run the numbers?

Why: Sum and difference formulas expand sin(A+B), cos(A-B), etc. — essential for simplifying expressions, solving equations, and deriving other identities.

How: sin(A+B)=sinAcosB+cosAsinB (note the + in the middle). cos(A+B)=cosAcosB-sinAsinB (note the -). For difference, flip the sign of the second term.

Double angle: set A=B to get sin(2A)=2sinAcosA, cos(2A)=cos²A-sin²A.sin(A-B)=sinAcosB-cosAsinB — the minus sign applies to the cosAsinB term.
Sources:Wolfram MathWorld — Trigonometric Addition FormulasKhan Academy — Trig Identities

Run the calculator when you are ready.

Start CalculatingEnter angles A and B to compute sin(A±B), cos(A±B), tan(A±B)

Examples — Click to Load

sum-diff.sh
CALCULATED
$ sum-diff --angles 30,45 --op sum --fn sin
sin(A)
0.5
sin(B)
0.70710678
A+B
75°
sin(A+B)
0.96592583
sin(A)
0.5
cos(A)
0.8660254
sin(B)
0.70710678
Identity
✓
Share:
Sum/Difference Calculator Result
sin(30+45°)
0.96592583
A=30° B=45°Identity: verified
numbervibe.com/calculators/mathematics/trigonometry/sum-difference-calculator

Value Breakdown

Original vs Combined

Component Contributions

Calculation Breakdown

INPUT/CONVERSION
Angle A
30°
Angle B
45°
Combined A + B
75°
A + B
FORMULA APPLICATION
sin(A), cos(A), tan(A)
0.5, 0.8660254, 0.57735027
sin(B), cos(B), tan(B)
0.70710678, 0.70710678, 1
Formula
sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
ext{Identity} ext{used}
PRIMARY RESULTS
SIN(A+B) RESULT
0.96592583
ext{Direct} ext{computation}
Formula gives
0.96592583
ext{From} ext{identity}
IDENTITY VERIFICATION
Identity verified
✓ Match
| ext{direct} - ext{formula}| < ε

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

✓

sin(A+B)=sinAcosB+cosAsinB — the sum formula. Difference: sin(A-B)=sinAcosB-cosAsinB.

— Wolfram MathWorld

2θ

Double angle formulas are the special case A=B of the sum formulas.

— Khan Academy

Key Takeaways

  • • sin(A±B) = sinAcosB ± cosAsinB — sine uses the same sign as the operation (sum → +, difference → −)
  • • cos(A±B) = cosAcosB ∓ sinAsinB — cosine flips: sum has minus, difference has plus between product terms
  • • tan(A±B) = (tanA ± tanB)/(1 ∓ tanAtanB) — denominator sign flips: sum uses 1−tanAtanB, difference uses 1+tanAtanB
  • • These identities derive from Euler's formula: e^(iA)·e^(iB) = e^(i(A+B)); expand and equate real/imaginary parts
  • • Used in wave superposition, rotation matrices, trigonometric substitution, and deriving double-angle and half-angle formulas

Did You Know?

📐Euler's formula e^(ix)=cos(x)+i·sin(x) yields the sum formulas: multiply e^(iA)·e^(iB) and equate real and imaginary partsSource: MIT OCW
🌊Wave superposition: sin(ωt+φ₁)+sin(ωt+φ₂) uses sum formulas to combine phases — fundamental in acoustics and opticsSource: Khan Academy
🔄Rotation matrices in 2D: R(α+β)=R(α)R(β) expand via cos(α+β) and sin(α+β) formulasSource: Wolfram MathWorld
∫Trigonometric substitution in calculus: ∫√(a²−x²)dx uses x=a·sinθ; sum formulas help with compound anglesSource: Paul's Notes
📡Signal processing: sum/difference identities explain frequency mixing and heterodyning in radioSource: IEEE DSP
🎯Double-angle formulas are A=B special cases: sin(2A)=2sinAcosA comes from sin(A+A)Source: NIST DLMF

How Sum and Difference Formulas Work

Sum and difference identities express trig functions of A±B in terms of trig functions of A and B. They are foundational—double-angle, half-angle, and product-to-sum all derive from them.

Euler Derivation

e^(i(A+B)) = e^(iA)·e^(iB) = (cosA+i·sinA)(cosB+i·sinB). Expanding: cos(A+B) + i·sin(A+B) = (cosAcosB−sinAsinB) + i(sinAcosB+cosAsinB). Equate real and imaginary parts to get both formulas at once.

Cosine Sign Rule

For cosine: sum has minus (cos(A+B)=cosAcosB−sinAsinB), difference has plus (cos(A−B)=cosAcosB+sinAsinB). Replace B with −B in the sum formula: cos(−B)=cosB, sin(−B)=−sinB, so the sin·sin term flips sign.

Tangent Formula

tan(A±B) = sin(A±B)/cos(A±B). Substitute the sine and cosine formulas, then divide numerator and denominator by cosAcosB to get (tanA±tanB)/(1∓tanAtanB). Undefined when denominator is zero.

Expert Tips

Cosine: Sum Minus, Diff Plus

Memorize: cos(A+B) has a minus between the product terms; cos(A−B) has a plus. The Double Angle Calculator uses A=B.

Sine: Same Structure

sin(A±B) = sinAcosB ± cosAsinB — the ± matches the operation. Sum → plus, difference → minus. See Product to Sum for the reverse.

Tangent Denominator Flips

tan(A+B) uses 1−tanAtanB; tan(A−B) uses 1+tanAtanB. When tanAtanB=1, tan(A+B) is undefined (A+B=90°+n·180°).

Exact Values

Use sum formulas to find exact values: sin(75°)=sin(45°+30°)=sin45°cos30°+cos45°sin30°=√2/2·√3/2+√2/2·1/2. Try the Sine Calculator.

Sum/Difference vs Other Methods

FeatureThis CalculatorDirect ComputeManual Formula
sin, cos, tan all supported✅✅✅
Two angle inputs A, B✅✅✅
Identity verification✅❌⚠️ Manual
Component breakdown✅❌✅
Charts & visualization✅❌❌
Step-by-step explanation✅❌✅
Degrees and radians✅✅✅
7 preset examples✅❌❌

Frequently Asked Questions

How do sum formulas relate to double-angle?

Set A = B. Then sin(A+B) = sin(2A) = sinAcosA + cosAsinA = 2sinAcosA. Similarly, cos(2A) = cos²A − sin²A. Double-angle is the special case of sum formulas.

Why does cosine have opposite signs for sum vs difference?

cos(A−B) = cos(A+(−B)) = cosAcos(−B) − sinAsin(−B) = cosAcosB + sinAsinB. The minus in the formula becomes plus because sin(−B) = −sinB.

Where are these used in practice?

Wave superposition (physics), rotation matrices (graphics, robotics), trigonometric substitution (calculus), signal processing (frequency mixing), and deriving other identities.

When is tan(A+B) undefined?

When 1 − tanAtanB = 0, i.e. tanAtanB = 1. This occurs when A+B = 90° + n·180° (e.g. A=30°, B=60° gives tan(90°) undefined).

How do I derive these from Euler's formula?

e^(i(A+B)) = e^(iA)e^(iB). Expand both sides using e^(ix)=cos(x)+i·sin(x). Equate real parts for cosine formula, imaginary for sine formula.

What is sin(15°) exactly?

sin(15°) = sin(45°−30°) = sin45°cos30° − cos45°sin30° = (√2/2)(√3/2) − (√2/2)(1/2) = √2(√3−1)/4.

Can I use these for any angles?

Yes. The formulas work for all real A and B. For tan(A±B), avoid angles where the denominator 1∓tanAtanB is zero.

How do product-to-sum formulas relate?

Product-to-sum is the reverse: they convert sinAsinB, cosAcosB, sinAcosB into sums. Add/subtract the sum and difference formulas to derive them.

Sum/Difference by the Numbers

6
Main Formulas
2
Angles A, B
±
Sum or Diff
e^(ix)
Euler Derivation

Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. For mission-critical applications (aerospace, medical devices), always verify with certified computational tools. Not a substitute for professional engineering analysis.

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