Sum and Difference Formula Calculator

Enter two angles A and B in degrees to evaluate all six sum and difference trigonometric identities instantly.

Created by Luis GonzalezLast updated: May 9, 2026

How to Use This Calculator

1
Enter Angle A in Degrees
Input the first angle for your trigonometric calculation. This can be any real number, including negative values or those exceeding 360 degrees.
2
Enter Angle B in Degrees
Input the second angle. The calculator will use this angle to compute the sum (A+B) and difference (A−B) identities.
3
View All Sum and Difference Results
The calculator will instantly display the values for sin(A±B), cos(A±B), and tan(A±B), along with their quadrant and sign analysis.

Example Calculation

A student needs to find the exact value of sin(45°) using the sum formula, by combining two known angles.

Angle A (°)

Angle B (°)

Results

0.70710678

Tips

Verify Special Angles

Use the calculator to confirm values for common special angles (e.g., 30°, 45°, 60°, 90°) by expressing them as sums or differences of other known angles. For instance, sin(75°) = sin(45° + 30°).

Understand Quadrant Signs

Pay attention to the quadrant each sum or difference angle falls into, as this determines the sign of the trigonometric function. 'All Students Take Calculus' (ASTC) is a useful mnemonic for remembering which functions are positive in each quadrant.

Tangent Identity Undefined

Remember that tan(θ) is undefined when θ is 90°, 270°, or any odd multiple of 90°. If A+B or A-B results in such an angle, the tangent value will not be finite.

The Sum and Difference Formula Calculator provides instant computations for trigonometric identities like sin(A±B), cos(A±B), and tan(A±B). By simply inputting two angles in degrees, this tool allows students, engineers, and physicists to explore fundamental trigonometric relationships. It is invaluable for simplifying complex expressions, finding exact values for non-standard angles (e.g., 75° or 105°), and understanding the behavior of oscillating systems in fields ranging from signal processing to celestial mechanics.

The Sum and Difference Identities Explained

The sum and difference formulas are foundational identities in trigonometry that express the sine, cosine, or tangent of a sum or difference of two angles in terms of the sines and cosines of the individual angles. These identities are derived from geometric principles, often using the unit circle or coordinate geometry, and are crucial for expanding and simplifying trigonometric expressions.

The 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)
tan(A - B) = (tan A - tan B) / (1 + tan A tan B)

Each variable A and B represents an angle, allowing for the decomposition of complex angles into simpler, known components.

💡 Understanding these fundamental angle relationships is key to many engineering problems. For example, applying trigonometric principles to rotational motion can be explored with our RPM Calculator (from SFM & Diameter).

Calculating sin(45°) Using Angle Sums

Let's use the calculator to find the value of sin(45°) by expressing it as the sum of two special angles, 30° and 15°. This demonstrates how to apply the sum formula for sine.

Input Angle A: 30°
Input Angle B: 15°
Compute A+B: 30° + 15° = 45°.
Apply sin(A+B) formula: sin(30° + 15°) = sin(30°)cos(15°) + cos(30°)sin(15°) = (0.5)(0.9659258) + (0.8660254)(0.2588190) = 0.4829629 + 0.2241438 = 0.7071067

The calculator will show that sin(A+B) for A=30° and B=15° is approximately 0.70710678, which is indeed the exact value of sin(45°) (√2/2).

💡 To delve into other mathematical properties of numbers, sometimes even seemingly "sad" ones, our Sad Number Checker offers a different kind of numerical exploration.

Expert Interpretation of Trigonometric Identities in Engineering

Engineers frequently utilize sum and difference formulas in fields such as electrical engineering, mechanical engineering, and signal processing. In electrical engineering, these identities are crucial for analyzing alternating current (AC) circuits, particularly when dealing with phase shifts and the superposition of sinusoidal waveforms. For example, when two AC signals with different phase angles combine, sum-to-product or product-to-sum identities (derived from these basic formulas) can simplify the resulting complex waveform, allowing engineers to predict amplitude and frequency characteristics. Mechanical engineers use them in vibration analysis, understanding how different oscillatory forces interact. Professionals look for patterns that indicate constructive or destructive interference, phase synchronization, or frequency modulation, all of which are directly modeled by these trigonometric relationships. The signs and magnitudes of the results help engineers determine system stability, resonance, and overall performance.

The Origins of Angle Sum and Difference Formulas

The origins of the angle sum and difference formulas can be traced back to ancient Greek mathematics, particularly with Ptolemy in the 2nd century AD. In his monumental work, the Almagest, Ptolemy developed a comprehensive system of trigonometry for astronomical calculations. He used chords of circles to define trigonometric quantities, effectively pre-dating our modern sine and cosine functions. Ptolemy's theorem on cyclic quadrilaterals and his chord subtraction and addition formulas are direct geometric precursors to what we now know as the sum and difference identities. For instance, his work on the chord of the difference of two arcs is equivalent to the modern sine difference formula. These initial developments were further elaborated and transformed into their modern algebraic forms by medieval Islamic mathematicians, such as Al-Battani and Abu al-Wafa, and later by European mathematicians like Regiomontanus and Euler, solidifying their place as cornerstones of trigonometry.

Frequently Asked Questions

What are the sum and difference formulas in trigonometry?

The sum and difference formulas are identities used to expand trigonometric functions of sums or differences of angles. For example, sin(A+B) = sin A cos B + cos A sin B, and cos(A-B) = cos A cos B + sin A sin B. These formulas are fundamental for simplifying expressions, solving equations, and deriving other identities in trigonometry and calculus.

When are sum and difference formulas typically used?

These formulas are used in mathematics and engineering to find exact trigonometric values for angles that are not standard (e.g., 75° or 15°) by expressing them as sums or differences of known special angles (e.g., 45° + 30°). They are also crucial for simplifying complex trigonometric expressions, solving trigonometric equations, and in applications involving wave interference and phase shifts in physics and electrical engineering.

How do sum and difference formulas relate to wave phenomena?

In physics, particularly with wave phenomena like sound or light, sum and difference formulas are essential for analyzing the superposition of waves. They help describe how two waves with different frequencies or phases combine, leading to phenomena such as beats, interference patterns, and amplitude modulation. These identities allow engineers to predict the resulting waveform's characteristics.

Can these formulas be used with angles outside 0 to 360 degrees?

Yes, the sum and difference formulas apply to any real-valued angles, including those outside the 0 to 360-degree range, as well as negative angles. Trigonometric functions are periodic, meaning their values repeat every 360 degrees (or 2π radians). The calculator automatically handles these periodic properties, ensuring accurate results regardless of the input angle magnitude.