Where They Break
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 8 of 52.
tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x) both divide by cos x, so both break wherever cos x is 0. csc(x) and cot(x) divide by sin x, so they break where the point is on the x-axis, at 0° and 180°.
On the x-axis sin x is 0, so csc x and cot x have no value.
This is why an identity is stated "wherever both sides are defined". tan(x)cos(x) = sin(x) holds at every angle except where cos x is 0. At those angles the right side has a value and the left side does not, so the identity makes no claim there.
Challenge
Tap every angle in one turn where tan(θ) and sec(θ) have no value. 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