continuity of sine and cosine

Theorem.  The real functions  xsinx  and  xcosx  are continuous at every real number x.

Proof.  Let ε be an arbitrary positive number.  Denote  Δsinx=:sinz-sinx,  Δcosx=:cosz-cosx  where we suppose that  |z-x|<π2.  We may interpret |z-x| as an arc of the unit circleMathworldPlanetmath of the xy-plane.  Let’s think in the circle the right triangleMathworldPlanetmath with hypotenuseMathworldPlanetmath the chord of the arc and the catheti (i.e. the shorter sides) vertical and horizontal.  Then |Δsinx| and |Δcosx| are just these cathets; so we have


If we make  |z-x|<ε,  then also  |Δsinx| and |Δcosx| are less than ε.  It means that both functions are continuous at x.


  • 1 E. Lindelöf: Johdatus korkeampaan analyysiin. Neljäs painos.  Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
Title continuity of sine and cosine
Canonical name ContinuityOfSineAndCosine
Date of creation 2013-03-22 19:15:37
Last modified on 2013-03-22 19:15:37
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 4
Author pahio (2872)
Entry type Theorem
Classification msc 26A15