Sum Difference Identities
Free online sum difference identities calculator computes sin(a+/-b), cos(a+/-b), and tan(a+/-b) for any two angles. Interactive graph for visualizing trigonometric sum and difference formulas.
About This Calculator
This sum and difference identities calculator evaluates all six sum/difference trigonometric identities -- sin(a+b), sin(a-b), cos(a+b), cos(a-b), tan(a+b), and tan(a-b) -- for any two input angles a and b in degrees. These identities are among the most fundamental in trigonometry, forming the basis for double-angle, half-angle, power-reducing, and product-to-sum formulas used throughout precalculus, calculus, and engineering.
The calculator implements the three core formulas: sin(a+/-b) = sin a cos b +/- cos a sin b, cos(a+/-b) = cos a cos b ∓ sin a sin b, and tan(a+/-b) = (tan a +/- tan b)/(1 ∓ tan a tan b). The interactive chart plots the identities as angle a varies from 0 deg to 360 deg while keeping angle b fixed, showing how the sum and difference functions compare across the full sine and cosine cycle.
Example
To find sin(75 deg), decompose 75 deg into 45 deg + 30 deg -- two angles with known exact trig values. Using the sine sum identity: sin(75 deg) = sin(45 deg+30 deg) = sin45 cos30 + cos45 sin30 = (sqrt2/2)(sqrt3/2) + (sqrt2/2)(1/2) = (sqrt6+sqrt2)/4 ≈ 0.9659. Enter a = 45 and b = 30 into the calculator to verify the result and explore all six identities at once. The chart mode lets you toggle between viewing sine curves, cosine curves, or all four together.
How to use
Enter angle a and angle b in degrees, then click Calculate. Six result cards appear showing all sum and difference values. The chart toggles between viewing all four sine/cosine curves, just the sine pair, or just the cosine pair. Values can be shared via the automatically updated URL, making it easy to bookmark or share specific angle combinations with classmates or colleagues.
Frequently Asked Questions
What are the sum and difference identities?
Sum and difference identities express trigonometric functions of a +/- b in terms of functions of a and b. For sine: sin(a+/-b) = sin(a)cos(b) +/- cos(a)sin(b). For cosine: cos(a+/-b) = cos(a)cos(b) ∓ sin(a)sin(b). For tangent: tan(a+/-b) = (tan(a) +/- tan(b))/(1 ∓ tan(a)tan(b)).
How do you derive the sine sum formula?
The sine sum formula sin(a+b) = sin(a)cos(b) + cos(a)sin(b) can be derived geometrically using the unit circle, or algebraically from Euler's formula: e^(i(a+b)) = e^(ia) x e^(ib) = (cos a + i sin a)(cos b + i sin b) = cos(a+b) + i sin(a+b). Expanding and equating imaginary parts gives the formula.
What is the mnemonic for sum and difference formulas?
A common mnemonic is 'Sine is sexy, Cosine is cosy'. For sine sum: sin(a+b) = sin(a)cos(b) + cos(a)sin(b) -- the signs stay the same (both +). For cosine sum: cos(a+b) = cos(a)cos(b) - sin(a)sin(b) -- the signs are opposite. For difference, the signs flip: sin(a-b) = sin(a)cos(b) - cos(a)sin(b) and cos(a-b) = cos(a)cos(b) + sin(a)sin(b).
How do sum and difference identities relate to double-angle formulas?
Double-angle formulas are a special case of sum identities where a = b. For example, setting a = b in sin(a+b) = sin(a)cos(b) + cos(a)sin(b) gives sin(2a) = 2 sin(a)cos(a). Similarly, cos(2a) = cos^2(a) - sin^2(a) from the cosine sum formula, and tan(2a) = 2 tan(a)/(1 - tan^2(a)) from the tangent sum formula.
Can sum and difference identities be used with any angle?
Yes, sum and difference identities work for any real angles, including negative angles and angles beyond 360 deg. They are particularly useful for finding exact values of trig functions at non-standard angles, such as sin(75 deg) = sin(45 deg+30 deg) = sin45 cos30 + cos45 sin30 = (sqrt6+sqrt2)/4.
How are sum and difference identities applied in physics?
In physics, these identities are used in wave superposition to combine waves of different phases and frequencies. For example, the sum of two sine waves sin(omegat) + sin(omegat + δ) simplifies using the sum identity to 2 cos(δ/2) sin(omegat + δ/2), showing how interference patterns depend on the phase difference δ.
What is the difference between sum identities and product-to-sum formulas?
Sum identities expand sin(a+/-b) and cos(a+/-b) into products of trig functions. Product-to-sum formulas reverse the process, rewriting products like sin(a)cos(b) as sums (1/2)[sin(a+b) + sin(a-b)]. While sum identities break compound angles apart, product-to-sum formulas are used for integrating products of trig functions and simplifying expressions in calculus.
How do you find exact trig values using sum identities?
To find exact values, decompose a non-standard angle into a sum or difference of known angles (30 deg, 45 deg, 60 deg). For example, sin(15 deg) = sin(45 deg - 30 deg) = sin45 cos30 - cos45 sin30 = (sqrt2/2)(sqrt3/2) - (sqrt2/2)(1/2) = (sqrt6 - sqrt2)/4. This method works for any angle expressible as a combination of the standard unit-circle angles.