Sum and Difference Identities Calculator
Our free trigonometry calculator solves sum difference identities problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy