Products to Sums
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 43 of 52.
Add sin(A + B) and sin(A - B): the cos A sin B terms cancel and two equal products are left. That, and its two cousins, turn a product into a sum:
2sin(A)cos(B) = sin(A + B) + sin(A - B)
2cos(A)cos(B) = cos(A - B) + cos(A + B)
2sin(A)sin(B) = cos(A - B) - cos(A + B)
With A = 3x and B = x, 2sin(3x)cos(x) is sin(4x) + sin(2x). A sum is easier to graph, to add up, and in calculus to integrate.
Challenge
Rewrite 2cos(5x)cos(3x) as a sum of two cosines.
Type each next line yourself; a line that does not equal the one above is refused.
Use the formula chip, then type the sum with each angle 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