Interactive Sum and Difference Identities Calculator

What Are Sum and Difference Identities?

Breaking down angle addition and subtraction formulas.
The six fundamental identities for sine, cosine, and tangent.
Why these identities are a cornerstone of trigonometry.

Sum and difference identities, also known as angle addition/subtraction formulas, are a set of trigonometric equations that express the trig function of a sum or difference of two angles (e.g., A + B or A - B) in terms of the trig functions of the individual angles A and B. They are a fundamental tool for simplifying complex expressions and finding exact trigonometric values for a wide range of angles.

The Six Core Formulas

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))

Core Identity Examples

Finding sin(75°) by calculating sin(45° + 30°).
Simplifying cos(x)cos(y) + sin(x)sin(y) to cos(x - y).

Step-by-Step Guide to Using the Calculator

Selecting the right function and operation.
Entering angles in degrees or radians.
Interpreting the calculated result and formula.

Our calculator simplifies the process of applying these identities. Follow these steps for an accurate calculation:

Input Guidelines

1. Select Trigonometric Function: Choose sine (sin), cosine (cos), or tangent (tan) from the dropdown menu.

2. Select Operation: Choose whether you want to calculate the sum (A + B) or the difference (A - B).

3. Enter Angles: Input the values for Angle A and Angle B.

4. Choose Unit: Specify whether your angles are in 'Degrees' or 'Radians'. This is critical for accuracy.

5. Calculate: Click the 'Calculate' button to see the result.

Understanding the Output

The calculator provides not just the final numerical answer, but also the specific formula used and a step-by-step breakdown of the intermediate values, helping you understand how the result was derived.

Practical Usage Examples

Input: func=cos, op=difference, A=60, B=45, unit=degrees → Result: cos(15°)
Input: func=tan, op=sum, A=π/6, B=π/4, unit=radians → Result: tan(5π/12)

Real-World Applications of Sum and Difference Identities

Analyzing wave interference in physics.
Simplifying circuit analysis in electrical engineering.
Creating rotations and transformations in computer graphics.

These identities are not just academic exercises; they are critical in many scientific and engineering fields.

Physics and Wave Mechanics

In physics, the superposition of waves (like sound, light, or water waves) is analyzed using sum and difference identities. When two waves meet, their resulting amplitude is determined by formulas derived directly from these identities, explaining phenomena like constructive and destructive interference.

Electrical Engineering

In AC circuit analysis, the phase relationships between voltage and current are described by sinusoidal functions. Sum and difference identities are used to analyze circuits with multiple AC sources, simplifying complex impedance calculations.

Navigation and Geodesy

These formulas are used in spherical trigonometry for calculations in navigation and geodesy, helping to determine distances and bearings on the Earth's surface.

Industry Applications

Calculating the beat frequency between two sound waves.
Analyzing the phase shift in an RLC circuit.
Determining the position of a satellite.

Common Misconceptions and Correct Methods

Avoiding the temptation to distribute functions.
Watching for sign errors in the cosine formulas.
Handling undefined tangent values correctly.

A common mistake is to incorrectly 'distribute' a trigonometric function over a sum or difference. It is crucial to use the correct identity.

Incorrect vs. Correct Method

Incorrect: sin(A + B) = sin(A) + sin(B)

Correct: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

Cosine Sign Rule

Remember the unique sign change in the cosine identities: cos(A + B) has a minus sign in its formula, while cos(A - B) has a plus sign. This is a frequent source of errors.

Tangent Edge Cases

The tangent function is undefined at 90° (π/2 radians) and its multiples. When using the tangent identities, be aware that if A, B, or A±B results in such an angle, the identity may involve division by zero and will be undefined. Our calculator handles these cases automatically.

Common Errors to Avoid

Mistaking cos(A+B) for cos(A)cos(B) + sin(A)sin(B).
Forgetting to check if tan(A) or tan(B) is undefined.

Mathematical Derivation and Proofs

Geometric proof using the unit circle.
Derivation via Euler's formula (e^ix = cos(x) + i*sin(x)).
Using vector dot products for a concise proof.

The identities can be elegantly proven through several methods, each providing a different insight.

Geometric Unit Circle Proof

One of the most intuitive proofs involves the unit circle. By constructing points P(A) and P(B) on the circle, we can use the distance formula between them and rotate the entire system by -B to place P(B) at (1,0). The distance remains the same, and equating the two expressions for the squared distance yields the identity for cos(A - B). Other identities can be derived from this starting point.

Euler's Formula Proof

A more advanced proof uses Euler's formula, e^(ix) = cos(x) + isin(x). By considering e^(i(A+B)) = e^(iA) e^(iB), we can expand both sides into their cosine and sine components. Equating the real and imaginary parts on both sides of the equation simultaneously proves the sum identities for both sine and cosine.

Proof Techniques

Distance formula proof for cos(A-B).
Expanding e^(i(A+B)) to derive sin and cos sum formulas.

Sine of Sum (sin(45° + 30°))

Cosine of Difference (cos(60° - 45°))

Tangent of Sum in Radians (tan(π/6 + π/4))

Sine of Difference with Negative Angle