Read the Formula Backwards
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 29 of 52.
Read from right to left, the formulas collapse a sum of products into one function. That finds exact values for angles that are not special themselves.
Take cos(80°)cos(20°) + sin(80°)sin(20°). Neither 80° nor 20° has an exact value you know, but the shape is cos A cos B + sin A sin B, the right side of cos(A - B). So it equals cos(80° - 20°), which is cos(60°), and that is 1/2.
Match the shape before the numbers: which functions pair up, and which sign joins them. cos cos with a plus is cos(A - B); cos cos with a minus is cos(A + B); sin cos plus cos sin is sin(A + B).
Challenge
Find the exact value of sin(50°)cos(10°) + cos(50°)sin(10°).
Type each next line yourself; a line that does not equal the one above is refused.
Fold it into one sine with the formula chip, type the sine of the worked-out angle, then its exact value. 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 Angles3Pythagorean Identities
1 + tan²x = sec²x1 + cot²x = csc²xA Difference of SquaresSquaring a SumRecap: Three Forms6Sum and Difference
sin(A + B) and sin(A - B)cos(A + B) and cos(A - B)Turning by 90° and 180°Read the Formula Backwardstan(A + B) and tan(A - B)Recap: Sum Formulas