Prosthaphaeresis Formulas
Prosthaphaeresis formulas convert products of trig functions into sums.
Learn these historical formulas with step-by-step examples.
The Formulas
sin(A) × sin(B) = ½[cos(A-B) - cos(A+B)]
sin(A) × cos(B) = ½[sin(A+B) + sin(A-B)]
cos(A) × sin(B) = ½[sin(A+B) - sin(A-B)]
Prosthaphaeresis (from the Greek words for "addition" and "subtraction") is a technique that converts the multiplication of two trigonometric values into addition and subtraction. Before the invention of logarithms, this was the primary method used for performing large multiplications.
The technique was developed in the late 16th century by astronomers who needed to multiply large numbers for celestial calculations. Johannes Werner described the method around 1510, and it was later refined by Tycho Brahe and his assistant Paul Wittich around 1580 in Denmark. The basic idea: to multiply two numbers, express them as cosines, apply the formula, and convert back.
Today these formulas remain important in signal processing, where they describe the phenomenon of beats (two frequencies combining to produce amplitude modulation). They also appear in Fourier analysis and the study of wave interference.
Variables
| Symbol | Meaning |
|---|---|
| A | First angle (in degrees or radians) |
| B | Second angle (in degrees or radians) |
| sin, cos | Standard trigonometric functions |
Example 1
Calculate cos(75°) × cos(15°) using prosthaphaeresis.
Apply: cos(A)cos(B) = ½[cos(A-B) + cos(A+B)]
= ½[cos(75°-15°) + cos(75°+15°)]
= ½[cos(60°) + cos(90°)]
= ½[0.5 + 0] = ½ × 0.5
cos(75°) × cos(15°) = 0.25
Example 2
Express sin(50°) × cos(20°) as a sum.
Apply: sin(A)cos(B) = ½[sin(A+B) + sin(A-B)]
= ½[sin(50°+20°) + sin(50°-20°)]
= ½[sin(70°) + sin(30°)]
= ½[0.9397 + 0.5] = ½ × 1.4397
sin(50°) × cos(20°) ≈ 0.7199
When to Use It
Prosthaphaeresis formulas are used in both pure mathematics and applied sciences.
- Simplifying products of trigonometric functions in calculus
- Signal processing — analyzing beat frequencies and modulation
- Physics — understanding wave interference patterns
- Integrating products of sine and cosine functions