Calculating sin' und cos' without power series methods


  1. The limit calculation

    lim x0 sinxsin0 x0 = lim x0 sinx x =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIWaaabeaakmaalaaabaGaci4CaiaacMgacaGGUbGaamiEaiabgkHiTiGacohacaGGPbGaaiOBaiaaicdaaeaacaWG4bGaeyOeI0IaaGimaaaacqGH9aqpdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamiEaiabgkziUkaaicdaaeqaaOWaaSaaaeaaciGGZbGaaiyAaiaac6gacaWG4baabaGaamiEaaaacqGH9aqpcaaIXaaaaa@560B@

    in [6.8.6] proves in fact the differentiability of sin at 0 with sin (0)=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaai4jaiaacIcacaaIWaGaaiykaiabg2da9iaaigdaaaa@3D43@ .
     

  2. For x[ π 2 , π 2 ] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaacUfacqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaacaGGDbaaaa@40DC@ we use Pythagoras' theorem (see [4.3.*]) to get cosx= 1 sin 2 (x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaamiEaiabg2da9maakaaabaGaaGymaiabgkHiTiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiaacIcacaWG4bGaaiykaaWcbeaaaaa@42A6@ , which means that cos and 1 sin 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaaIXaGaeyOeI0Iaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@3B65@ coincide locally at 0. As 1 sin 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgkHiTiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaaaaa@3B55@ is differentiable at 0 and X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacaWGybaaleqaaaaa@36E4@ at 1, the chain rule ([7.6.11]) proves cos to be differentiable at 0 with

    cos (0)= 2sin0sin (0) 2 1 sin 2 (0) =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaai4jaiaacIcacaaIWaGaaiykaiabg2da9maalaaabaGaeyOeI0IaaGOmaiGacohacaGGPbGaaiOBaiaaicdacqGHflY1ciGGZbGaaiyAaiaac6gacaGGNaGaaiikaiaaicdacaGGPaaabaGaaGOmamaakaaabaGaaGymaiabgkHiTiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiaacIcacaaIWaGaaiykaaWcbeaaaaGccqGH9aqpcaaIWaaaaa@53D5@ .

     
  3. Using the addition formulas for sine and cosine (see [4.3.*]) we get for all x,a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWGHbGaeyicI4SaeSyhHekaaa@3B73@ :

    sinx=sin(xa+a)=sin(xa)cosa+cos(xa)sina cosx=cos(xa+a)=cos(xa)cosasin(xa)sina MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiGacohacaGGPbGaaiOBaiaadIhacqGH9aqpciGGZbGaaiyAaiaac6gacaGGOaGaamiEaiabgkHiTiaadggacqGHRaWkcaWGHbGaaiykaiabg2da9iGacohacaGGPbGaaiOBaiaacIcacaWG4bGaeyOeI0IaamyyaiaacMcacqGHflY1ciGGJbGaai4BaiaacohacaWGHbGaey4kaSIaci4yaiaac+gacaGGZbGaaiikaiaadIhacqGHsislcaWGHbGaaiykaiabgwSixlGacohacaGGPbGaaiOBaiaadggaaeaaciGGJbGaai4BaiaacohacaWG4bGaeyypa0Jaci4yaiaac+gacaGGZbGaaiikaiaadIhacqGHsislcaWGHbGaey4kaSIaamyyaiaacMcacqGH9aqpciGGJbGaai4BaiaacohacaGGOaGaamiEaiabgkHiTiaadggacaGGPaGaeyyXICTaci4yaiaac+gacaGGZbGaamyyaiabgkHiTiGacohacaGGPbGaaiOBaiaacIcacaWG4bGaeyOeI0IaamyyaiaacMcacqGHflY1ciGGZbGaaiyAaiaac6gacaWGHbaaaaaa@8921@

    With the results of 1. and 2. we may apply the chain rule again. Thus, with a factor rule ([7.6.6]) argument, we find that sin and cos are differentiable at a with the following derivation numbers

    sin (a)=sin (0)1cosa+cos (0)1sina=cosa cos (a)=cos (0)1cosasin (0)1sina=sina MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8A3A@