Cosine of a difference calculator

Enter both angles in degrees or radians, and the result shows the sine and cosine of each angle, the two products, their sum, and the final value. This makes the calculator useful for getting a numerical answer and for checking every stage of textbook work without hiding the identity behind a single calculator key.

Set up the two angles in the correct order

Enter the first angle as A and the second angle as B, then select the unit that applies to both. The calculator evaluates cos(A − B), so it subtracts B from A before describing the difference. Although cosine is an even function and cos(A − B) equals cos(B − A), keeping the requested order is still important when you compare the output with a written exercise or reuse the displayed difference in later work. Degrees are the default and are convenient for familiar values such as 30, 45, 60, and 90. Choose radians when the angles come from calculus, programming, physics, or a formula already expressed in radians. Do not convert one angle while leaving the other unchanged: the selected unit is shared by both inputs. Negative angles and angles outside one full turn are accepted because the sine and cosine functions are periodic and remain defined for every finite real angle. The output repeats both inputs and their difference so you can verify the setup before trusting the arithmetic.

Follow the cosine difference identity term by term

The calculation uses the identity cos(A − B) = cos(A)cos(B) + sin(A)sin(B). First, it finds cosine A and cosine B and multiplies them to produce the cosine product. Next, it finds sine A and sine B and multiplies those values to produce the sine product. Finally, it adds the two products. The plus sign is the detail most often confused with the cosine sum identity: subtracting the angles inside cosine produces addition between the products, while adding the angles produces subtraction between them. The response exposes cosine_a, cosine_b, sine_a, sine_b, cosine_product, and sine_product before returning value, so each line of a manual solution can be checked independently. Results are rounded to twelve decimal places to suppress insignificant binary floating-point noise while retaining ample precision for ordinary numerical work. Values extremely close to zero are returned as zero, avoiding distracting results such as a tiny scientific-notation remainder where exact-angle mathematics expects zero.

Interpret and verify the returned value

The final value is the sum of the two displayed products and represents the cosine of the angle difference. Like every cosine value for a real angle, it should lie between −1 and 1, inclusive. A value near 1 means the two directions are close after periodicity is considered; a value near 0 means their difference is near a right angle; and a value near −1 means their directions are nearly opposite. This geometric interpretation makes the result useful beyond identity exercises, including comparing headings, resolving phase differences, checking rotations, and computing normalized directional similarity. For a quick independent check, subtract the input angles and evaluate the cosine of that single difference using the same unit. The direct result and the identity result should agree apart from normal rounding. The endpoint returns transparent numeric components rather than a symbolic proof, so it is best suited to supplied numerical angles. It does not parse expressions such as pi/6, infer mixed units, or simplify variables. API automation costs $0.002 per item, while the generated browser calculator uses the same deterministic solving function.