Sum and Difference Identities Calculator
Our free trigonometry calculator solves sum difference identities problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Sum and Difference Identities Calculator
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Formula: sin(A+B) = sinA cosB + cosA sinB | cos(A+B) = cosA cosB - sinA sinB
Worked example — sin(75) = 0.96592583 (identity verified against direct computation)
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) = (0.7071)(0.8660) + (0.7071)(0.5) = 0.6124 + 0.3536 = 0.9659 Direct: sin(75) = 0.9659 Exact: (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) = (0.7071)(0.8660) + (0.7071)(0.5) = 0.6124 + 0.3536 = 0.9659 Direct: cos(15) = 0.9659 Note: 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.
What are the sum-to-product identities?
Sum-to-product identities are the inverses of product-to-sum identities, converting sums of trigonometric functions into products. They are: sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2), sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2), cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2), and cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2). These are derived from the product-to-sum formulas by substituting A = (X+Y)/2 and B = (X-Y)/2. Sum-to-product identities are particularly useful for solving trigonometric equations like sin(3x) + sin(x) = 0 by converting the sum into a product 2 sin(2x) cos(x) = 0, which factors into simpler equations. They also explain the phenomenon of beats in acoustics, where two close frequencies combine to produce an amplitude-modulated wave.
How are sum and difference identities used in complex number multiplication?
When complex numbers are expressed in polar form as r(cos theta + i sin theta), multiplying two complex numbers z1 = r1(cos A + i sin A) and z2 = r2(cos B + i sin B) yields z1 z2 = r1 r2(cos(A+B) + i sin(A+B)). The real part cos(A+B) = cos A cos B - sin A sin B (the cosine sum identity) and the imaginary part sin(A+B) = sin A cos B + cos A sin B (the sine sum identity). This connection between complex multiplication and angle addition is formalized in Euler's formula e^(i theta) = cos theta + i sin theta, where e^(i(A+B)) = e^(iA) e^(iB) immediately gives both sum identities. This perspective reveals that sum and difference identities are fundamentally about the multiplicative structure of the unit circle in the complex plane.
How do engineers use sum and difference identities in signal processing?
Sum and difference identities are fundamental to signal processing and telecommunications. In radio communications, a mixer (multiplier) combines a signal sin(omega_s t) with a local oscillator cos(omega_lo t) to produce (1/2)[sin((omega_s + omega_lo)t) + sin((omega_s - omega_lo)t)], creating sum and difference frequencies. This is exactly the product-to-sum identity. AM radio uses this principle for modulation and demodulation. In digital signal processing, the discrete Fourier transform relies on complex exponentials that embody these identities. Phase detection circuits compute the phase difference between two signals using the product cos A cos B + sin A sin B = cos(A-B). Quadrature demodulation separates in-phase and quadrature components using these identities, enabling modern digital communications like Wi-Fi and cellular networks.
Can sum and difference identities be generalized to more than two angles?
Yes, sum and difference identities extend to three or more angles through recursive application. For three angles: sin(A + B + C) = sin A cos B cos C + cos A sin B cos C + cos A cos B sin C - sin A sin B sin C. This is obtained by writing sin((A+B) + C) and applying the two-angle identity twice. The general pattern for n angles involves sums of products where each term has an odd number of sine factors. For cosine of n angles, each term has an even number of sine factors (including zero). These generalized formulas connect to Chebyshev polynomials and the expansion of complex exponentials. In practice, recursive application of the two-angle formulas is more manageable than memorizing the multi-angle versions, and computational tools like Sum and Difference Identities Calculator make the actual numerical evaluation straightforward for any combination of angles.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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