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Sum And Difference Identities Calculator

Compute sin(A±B), cos(A±B), and tan(A±B) instantly using sum and difference identities. Enter two angles in degrees or radians for exact trigonometric results.

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Inputs

Identity Type

Angle A

Angle B

Angle Unit

Identity Value

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Identity Value—

The formula

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What Are Sum and Difference Identities?

Sum and difference identities — also called angle addition and subtraction formulas — are a foundational set of trigonometric equations that express the sine, cosine, and tangent of the sum or difference of two angles in terms of the trigonometric functions of each individual angle. These identities enable exact computation of trigonometric values for non-standard angles and appear throughout calculus, physics, electrical engineering, and computer graphics.

The Six Core Formulas

According to Trigonometric Identities and Equations (ScholarWorks@GVSU) and Texas A&M ODP Math 150, Chapter 8.2, the identities are stated as follows:

Sine Sum: sin(A + B) = sin A cos B + cos A sin B
Sine Difference: sin(A − B) = sin A cos B − cos A sin B
Cosine Sum: cos(A + B) = cos A cos B − sin A sin B
Cosine Difference: cos(A − B) = cos A cos B + sin A sin B
Tangent Sum: tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
Tangent Difference: tan(A − B) = (tan A − tan B) / (1 + tan A tan B)

Variable Definitions

A (Angle A) and B (Angle B) are the two input angles, each expressible in degrees or radians. The calculator converts inputs automatically before applying any formula. In the compact ± notation, the upper sign applies to the sum identity and the lower sign to the difference identity. The ∓ symbol in the cosine formula indicates the opposite sign from the selected ±.

Derivation of the Formulas

The sine sum formula has several proofs. The algebraic proof uses Euler's formula: e^i(A+B) = e^iA × e^iB. Expanding each factor as (cos + i·sin) and multiplying out, then separating real and imaginary parts, simultaneously yields cos(A + B) = cos A cos B − sin A sin B and sin(A + B) = sin A cos B + cos A sin B. The tangent identities follow by dividing sin(A+B) by cos(A+B) and simplifying, as demonstrated in OER Math 1060 – Trigonometry (Salt Lake Community College). A geometric proof using two right triangles inscribed within a rectangle produces the same result without complex numbers.

Worked Example 1 — sin(75°) Using the Sine Sum Identity

Express 75° = 45° + 30°. From the standard unit circle: sin(45°) = cos(45°) = √2/2, sin(30°) = 1/2, and cos(30°) = √3/2. Substituting into 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 ≈ 0.9659

This exact radical form is the standard expected answer in mathematics courses and cannot be obtained by pressing a decimal approximation button alone.

Worked Example 2 — cos(15°) Using the Cosine Difference Identity

Express 15° = 45° − 30°. Apply cos(A − B) = cos A cos B + sin A sin B:

cos(15°) = cos(45°)cos(30°) + sin(45°)sin(30°)
= (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4 ≈ 0.9659

The result matches sin(75°), confirming the co-function identity sin θ = cos(90° − θ) as a built-in consistency check.

Worked Example 3 — tan(15°) in Radians

Set A = π/3 radians (60°) and B = π/4 radians (45°). Apply the tangent difference formula: tan(π/3 − π/4) = (tan(π/3) − tan(π/4)) / (1 + tan(π/3)·tan(π/4)) = (√3 − 1)/(1 + √3) ≈ 0.2679, which equals the exact closed form 2 − √3, verifiable by rationalization.

Practical Applications

Calculus: The first-principles proof that d/dx[sin x] = cos x expands sin(x + h) using the sum identity, isolating cos(x) as h → 0.
AC Electrical Engineering: Phase-shifted voltages are combined using the cosine sum identity to determine resultant amplitude and phase angle.
Computer Graphics: 2D and 3D rotation matrices apply cos(A + B) and sin(A + B) to rotate vectors through compound angles without recomputing from scratch.
Signal Processing: Product-to-sum conversions — derived from the sum and difference identities — decompose amplitude-modulated signals into frequency components.
Navigation and GPS: Spherical trigonometry formulas used in great-circle routing build directly on planar sum and difference identities extended to spherical geometry.

How to Use This Calculator

Select the identity type (sine, cosine, or tangent), enter Angle A and Angle B, choose degrees or radians, and click Calculate. The tool applies the appropriate formula and returns the result as a decimal. Cross-check results by computing the equivalent combined angle directly on a unit circle or by verifying with a co-function relation. See Trigonometric Identities — Kennesaw State University for a concise reference sheet of all standard identities.

Reference

Frequently asked questions

What are sum and difference identities in trigonometry?

Sum and difference identities are six formulas — two each for sine, cosine, and tangent — that express the trigonometric function of the sum or difference of two angles using the trig functions of each angle individually. For example, sin(A + B) = sin A cos B + cos A sin B. These identities are essential for computing exact trig values for non-standard angles, simplifying complex expressions, and proving other identities throughout precalculus and calculus.

How do you use the sine sum identity to calculate sin(75 degrees)?

Express 75° as 45° + 30° and apply sin(A + B) = sin A cos B + cos A sin B. Substituting known unit-circle values: sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4, which is approximately 0.9659. This method yields an exact radical form — the expected answer in most math courses and standardized examinations — rather than a rounded decimal approximation.

Why is sin(A + B) not equal to sin A + sin B?

Sine is a nonlinear function, so it does not distribute over addition. The correct formula is sin(A + B) = sin A cos B + cos A sin B. A quick counterexample: sin(90° + 90°) = sin(180°) = 0, whereas sin(90°) + sin(90°) = 1 + 1 = 2. The two expressions differ by 2, confirming that the naive distributive assumption is false. Always apply the proper identity to avoid this widespread error in trigonometry.

Does the sum and difference identities calculator support radian input?

Yes. The calculator accepts angle inputs in either degrees or radians via the Angle Unit selector. After the unit is chosen, both Angle A and Angle B are interpreted in that unit, and the tool converts to radians internally before applying the formula. For example, entering A = π/4 rad and B = π/6 rad produces the same result as entering A = 45° and B = 30°, confirming unit consistency throughout the computation.

How is the tangent sum formula tan(A + B) derived from sine and cosine?

The tangent sum formula is derived by dividing the sine sum by the cosine sum: tan(A + B) = sin(A + B)/cos(A + B) = (sin A cos B + cos A sin B)/(cos A cos B − sin A sin B). Dividing every term in both numerator and denominator by cos A cos B yields (tan A + tan B)/(1 − tan A tan B). The formula is undefined when tan A tan B = 1, which occurs when A + B equals 90° plus any integer multiple of 180°, since cosine is zero at those angles.

What is a real-world example of applying the cosine difference identity?

In electrical engineering, an AC voltage signal is written as V(t) = V₀ cos(ωt − φ), where φ is the phase offset. Applying the cosine difference identity expands this as V₀[cos(ωt)cos(φ) + sin(ωt)sin(φ)], separating the signal into an in-phase component proportional to cos(ωt) and a quadrature component proportional to sin(ωt). This I/Q decomposition is the mathematical foundation of radio communications, radar signal processing, and modern software-defined radio systems.