Sums to Products
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 44 of 52.
Read the product formulas backwards, with the half-sum and the half-difference of the angles, and a sum becomes a product:
sin(A) + sin(B) = 2sin((A + B)/2)cos((A - B)/2)
cos(A) + cos(B) = 2cos((A + B)/2)cos((A - B)/2)
For sin(5x) + sin(x), the half-sum is 3x and the half-difference is 2x, so it is 2sin(3x)cos(2x).
This is how two notes of nearly the same pitch sound together. Their sum is a fast wave at the average pitch times a slow wave at half the difference in pitch. The loudness follows the size of the slow wave, which peaks twice in each of its cycles, so the beat you hear repeats at the full difference: notes of 440 Hz and 442 Hz beat twice a second.
Challenge
In a simple model, two notes played together give the wave sin(42t) + sin(38t). Rewrite it as a product of a sine and a cosine.
Type each next line yourself; a line that does not equal the one above is refused.
Use the formula chip, then type the product with the half-sum and half-difference worked out. Then press Check.
Try it yourself
Simplify the expression
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Trig Identities
1Identities on the Circle
sin²x + cos²x = 1Squares Erase the SignsIdentity or Equation?Recap: the Circle Identity4Even, Odd and Cofunctions
Negative AnglesEven and Odd in a ProofEven and Odd GraphsCofunctionsRecap: Signs of Angles2Reciprocal and Quotient
The Quotient IdentitySecant, Cosecant, CotangentDividing by a FractionWhere They BreakRecap: Reciprocals5Proving Identities
Work on One SideRewrite in sin and cosFactor FirstMake One FractionProofs That Go WrongRecap: Prove It8Half Angles, Power Reduction
Power Reductionsin²x Is a WaveHalf-Angle Valuestan of a Half AngleRecap: Half Angles