Exact Values, Doubled
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 34 of 52.
Read backwards, the double-angle formulas turn an angle you do not know into its double, which you may know. Take 2sin(15°)cos(15°). Its shape is 2 sin A cos A, so it is sin(2 · 15°), which is sin(30°), and that is 1/2.
The same works with any form of cos 2x. A line that reads cos²A - sin²A, or 2cos²A - 1, or 1 - 2sin²A, is cos 2A; double the angle and read the value off the special angles.
Challenge
Find the exact value of cos(22.5°)² - sin(22.5°)².
On this board the formula chip writes the values unworked; type each next line yourself in the box, and a line that does not equal the one above is refused.
Fold it into one cosine with the formula chip, type the cosine of the doubled 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 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