Sum and Difference Identities Calculator
Our free trigonometry calculator solves sum difference identities problems. Get worked examples, visual aids, and downloadable results.
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Formula
The sum identities express sin and cos of combined angles using the individual angle values. The difference identities use subtraction: sin(A-B) = sinA cosB - cosA sinB, cos(A-B) = cosA cosB + sinA sinB. For tangent: tan(A+B) = (tanA + tanB) / (1 - tanA tanB).
Last reviewed: December 2025
Worked Examples
Example 1: Sum Identity: sin(45 + 30) = sin(75)
Example 2: Difference Identity: cos(45 - 30) = cos(15)
Background & Theory
The Sum and Difference Identities Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Sum and Difference Identities Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
sin(A+B) = sinA cosB + cosA sinB | cos(A+B) = cosA cosB - sinA sinB
The sum identities express sin and cos of combined angles using the individual angle values. The difference identities use subtraction: sin(A-B) = sinA cosB - cosA sinB, cos(A-B) = cosA cosB + sinA sinB. For tangent: tan(A+B) = (tanA + tanB) / (1 - tanA tanB).
Worked Examples
Example 1: Sum Identity: sin(45 + 30) = sin(75)
Problem: Use the sum identity to compute sin(75 degrees) from sin(45+30).
Solution: sin(45+30) = sin(45)cos(30) + cos(45)sin(30)\n= (0.7071)(0.8660) + (0.7071)(0.5)\n= 0.6124 + 0.3536\n= 0.9659\nDirect: sin(75) = 0.9659\nExact: (sqrt(6) + sqrt(2)) / 4
Result: sin(75) = 0.96592583 (identity verified against direct computation)
Example 2: Difference Identity: cos(45 - 30) = cos(15)
Problem: Use the difference identity to compute cos(15 degrees).
Solution: cos(45-30) = cos(45)cos(30) + sin(45)sin(30)\n= (0.7071)(0.8660) + (0.7071)(0.5)\n= 0.6124 + 0.3536\n= 0.9659\nDirect: cos(15) = 0.9659\nNote: sin(75) = cos(15) (cofunction identity)
Result: cos(15) = 0.96592583 (matches sin(75) by cofunction identity)
Frequently Asked Questions
What are the sum and difference identities for sine and cosine?
The sum and difference identities express the sine and cosine of a sum or difference of two angles in terms of the sines and cosines of the individual angles. Specifically: 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, and cos(A - B) = cos A cos B + sin A sin B. These four identities are among the most important in all of trigonometry, as many other identities (double angle, half angle, product-to-sum) can be derived from them. They enable computation of trigonometric values at non-standard angles by breaking them into standard angle components, such as sin(75) = sin(45 + 30).
How do you derive the sum identity for sine?
The sum identity sin(A + B) = sin A cos B + cos A sin B can be derived geometrically using the unit circle. Draw angle A from the positive x-axis, then angle B from the terminal side of A. The point on the unit circle at angle (A + B) has y-coordinate sin(A + B). By constructing perpendiculars and using the right triangles formed, you can show this y-coordinate equals sin A cos B + cos A sin B. An alternative algebraic proof uses the rotation matrix: rotating a point (cos A, sin A) by angle B uses the matrix [[cos B, -sin B], [sin B, cos B]], and the resulting y-coordinate is sin A cos B + cos A sin B. Euler's formula provides yet another proof: e^(i(A+B)) = e^(iA) e^(iB), and equating imaginary parts gives the identity.
What is the tangent sum identity and when is it undefined?
The tangent sum identity is tan(A + B) = (tan A + tan B) / (1 - tan A tan B). This is derived by dividing the sine sum identity by the cosine sum identity: sin(A+B)/cos(A+B) = (sin A cos B + cos A sin B)/(cos A cos B - sin A sin B), then dividing numerator and denominator by cos A cos B. The identity is undefined in two cases: when tan A or tan B is undefined (at odd multiples of 90 degrees), and when the denominator 1 - tan A tan B equals zero, which occurs when tan A tan B = 1 (meaning A + B is an odd multiple of 90 degrees, where tangent itself is undefined). The difference version is tan(A - B) = (tan A - tan B) / (1 + tan A tan B), with similar restrictions. These formulas are essential for adding angles in surveying and navigation.
How can sum and difference identities evaluate sin(75 degrees) exactly?
To find sin(75) exactly, decompose 75 as 45 + 30 (both standard angles). Apply the sum identity: sin(75) = sin(45 + 30) = sin 45 cos 30 + cos 45 sin 30 = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = sqrt(6)/4 + sqrt(2)/4 = (sqrt(6) + sqrt(2))/4, approximately 0.9659. Similarly, cos(15) = cos(45 - 30) = cos 45 cos 30 + sin 45 sin 30 = (sqrt(6) + sqrt(2))/4, confirming the cofunction identity sin(75) = cos(15). This technique works for any angle that can be expressed as a sum or difference of standard angles: 15 = 45 - 30, 75 = 45 + 30, 105 = 60 + 45, and so on. It demonstrates how four standard angle values generate exact results for infinitely many angles.
How do sum identities lead to double angle formulas?
Double angle formulas are a special case of sum identities where both angles are equal (B = A). Setting A = B in sin(A + B) gives sin(2A) = sin A cos A + cos A sin A = 2 sin A cos A. For cosine: cos(2A) = cos A cos A - sin A sin A = cos^2 A - sin^2 A. Using the Pythagorean identity, this can also be written as 2cos^2 A - 1 or 1 - 2sin^2 A. For tangent: tan(2A) = (tan A + tan A)/(1 - tan A tan A) = 2 tan A/(1 - tan^2 A). These double angle formulas are used extensively in calculus for integration, in physics for wave interference, and in engineering for signal processing. Triple angle formulas can similarly be derived by setting B = 2A and applying the identities recursively.
What are the product-to-sum identities and how are they related?
Product-to-sum identities convert products of trigonometric functions into sums, making them easier to integrate or analyze. They are: sin A cos B = (1/2)[sin(A+B) + sin(A-B)], cos A sin B = (1/2)[sin(A+B) - sin(A-B)], cos A cos B = (1/2)[cos(A-B) + cos(A+B)], and sin A sin B = (1/2)[cos(A-B) - cos(A+B)]. These are derived directly from the sum and difference identities by adding or subtracting pairs. For example, adding sin(A+B) = sinAcosB + cosAsinB and sin(A-B) = sinAcosB - cosAsinB gives 2 sinAcosB = sin(A+B) + sin(A-B). Product-to-sum identities are crucial in signal processing for understanding amplitude modulation, where multiplying a signal by a carrier creates sum and difference frequencies.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy