Sum and Difference Identities Calculator
Calculate exact trigonometric values using angle addition and subtraction formulas.
Select a trig function and operation, enter two angles, and get an exact result with the formula shown.
Sum and Difference Identities Calculator
Calculate exact trigonometric values using angle addition and subtraction formulas.
About Sum and Difference Identities
Sum and difference identities, also known as angle addition and subtraction formulas, are a set of trigonometric equations that express the trig function of a sum or difference of two angles in terms of the trig functions of the individual angles. They allow you to evaluate sin, cos, or tan of angles like 75° or 15° exactly, without a calculator, by splitting them into angles whose trig values you already know (such as 30°, 45°, and 60°).
The six core formulas are: 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)); and tan(A − B) = (tan(A) − tan(B)) / (1 + tan(A)tan(B)). These identities are exact — they involve no approximation — and they can be proved using the unit circle or geometric arguments.
The calculator accepts angles in both degrees and radians. When using degrees, enter values like 30, 45, or 90. When using radians, enter decimal approximations (such as 0.5236 for π/6) or exact decimal values. The choice of unit matters because the trigonometric functions compute different values for the same number depending on whether it represents degrees or radians.
In physics, these identities are indispensable for analyzing wave superposition. When two waves of the same frequency but different phases combine, the amplitude of the resulting wave is determined directly by sum-and-difference formulas. In electrical engineering, phase relationships between voltage and current in AC circuits are governed by the same mathematics. Computer graphics uses them to rotate points and vectors in 2D and 3D space — a rotation matrix is built entirely from cosine and sine of the rotation angle.
In mathematics, sum and difference identities are a gateway to deriving many other trigonometric relationships: the double-angle formulas, the half-angle formulas, and the product-to-sum formulas all follow directly from these six identities. Mastering them provides a foundation for calculus, Fourier analysis, signal processing, and any field that relies on periodic functions.
Sum and Difference Identities Examples
Real examples showing how to apply the identities for common angles.
| Input | Result | Notes |
|---|---|---|
| sin(45° + 30°) | sin(75°) ≈ 0.9659 | sin(45°)cos(30°) + cos(45°)sin(30°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6+√2)/4. |
| cos(60° − 45°) | cos(15°) ≈ 0.9659 | cos(60°)cos(45°) + sin(60°)sin(45°) = (1/2)(√2/2) + (√3/2)(√2/2) = (√2+√6)/4. |
| tan(π/6 + π/4) in radians | tan(5π/12) ≈ 3.7321 | (tan(π/6)+tan(π/4))/(1−tan(π/6)tan(π/4)) = (1/√3+1)/(1−1/√3) simplified. |
| sin(120° − (−30°)) | sin(150°) = 0.5 | Negative angle B: sin(150°) = sin(90°+60°) = 0.5. Demonstrates correct negative-angle handling. |
How to Use the Sum and Difference Identities Calculator
- Choose the trigonometric function (Sine, Cosine, or Tangent) from the first dropdown.
- Select the operation: Sum (A + B) or Difference (A − B) from the second dropdown.
- Enter Angle A and Angle B in the respective fields.
- Choose the angle unit (Degrees or Radians) to match your input.
- Click 'Calculate' to see the numerical result and the specific identity formula that was applied.
Sum and Difference Identities FAQ
What are sum and difference identities used for?
They let you compute exact trig values for angles formed by combining standard angles (like 30°, 45°, 60°), and they underpin many other trigonometric identities. In applied fields they are used in physics for wave analysis, in engineering for AC circuit phase calculations, and in computer graphics for rotations.
When should I use degrees vs. radians?
Use degrees for everyday geometry and navigation problems, and for working with angles from tables or protractors. Use radians for calculus, physics, and engineering, where the natural unit for angles in formulas is radians. The calculator uses whichever unit you select — just make sure Angle A and Angle B use the same unit.
What makes tangent different from sine and cosine for these formulas?
The tangent sum/difference formulas involve a denominator (1 ∓ tan(A)tan(B)) that can be zero, which makes tangent undefined at those angle combinations. The calculator detects this and shows an error. Sine and cosine sum/difference formulas have no such restriction and work for all real angles.
How can I find sin(75°) using these identities?
Write 75° = 45° + 30°, then apply sin(A+B) = sin(A)cos(B) + cos(A)sin(B). Substituting: sin(45°)cos(30°) + cos(45°)sin(30°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6+√2)/4 ≈ 0.9659. Enter Function=sin, Operation=Sum, A=45, B=30 in the calculator to verify.
Can these identities work with negative angles?
Yes. The identities hold for all real angles, positive or negative. For example, sin(A − B) where B is negative is the same as sin(A + |B|). The calculator accepts negative values in the angle fields and applies the formulas correctly.
Are the results exact or approximate?
The numerical results are floating-point approximations computed to about 15 significant digits. The formulas themselves are exact mathematical identities — for example, sin(75°) = (√6+√2)/4 exactly — but the calculator displays the decimal equivalent for practical use.