Identity or Equation?
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 3 of 52.
sin(x) = 1/2 is true at some angles and false at others: in one turn only at 30° and 150°. That is an equation, and solving it means finding those angles. sin(x)² + cos(x)² = 1 is true at every angle. That is an identity, and there is nothing to solve.
To show that a claim is not an identity, one angle where both sides have values that differ is enough. Take sin(x) + cos(x) = 1. At 0° it reads 0 + 1 = 1 and at 90° it reads 1 + 0 = 1, so two checks pass. At 60° it reads √(3)/2 + 1/2, about 1.37, so it is not an identity.
The same example shows why checking angles never proves an identity: the claim passed twice and is still false. A proof rewrites one side into the other with steps that hold at every angle.
Challenge
Where is sin(θ) = cos(θ) true? Those are the angles where the point is as far up as it is across, so its tangent is 1. Tap every such angle in one turn. Then press Check.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
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