sin 2x = 2 sin x cos x
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 32 of 52.
Put B = A into the sum formula: sin(A + A) = sin(A)cos(A) + cos(A)sin(A). The two products are the same, so
sin(2x) = 2sin(x)cos(x)
Doubling the angle does not double the sine: at 30°, sin(60°) is about 0.87, not 2sin(30°) = 1. The 2 comes out only together with the cos x.
Take sin(2x)/cos(x). Write the top as 2sin(x)cos(x); the bottom cos x is now a factor of the top, so it cancels and leaves 2sin(x), wherever cos x is not 0.
Challenge
Show that sin(2x)/sin(x) equals 2cos(x): rewrite it until the board reads 2cos(x). 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 Angles7Double-Angle Formulas
sin 2x = 2 sin x cos xThree Forms of cos 2xExact Values, DoubledSquare, Then DoubleThe Graph of 2 sin x cos xRecap: Double 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