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Lesson 10: Derivatives of Trigonometric Functions

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Lesson 10: Derivatives of Trigonometric Functions
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Section 3.4Derivatives of Trigonometric FunctionsMath 1aFebruary 25, 2008AnnouncementsGet 50% of all ALEKS points between now and 3/7Problem Sessions Sunday, Thursday, 7pm, SC 310Oﬃce hours Tuesday, Wednesday 2–4pm SC 323Midterm I Friday 2/29 in class (up to §3.2)Two important trigonometric limitsTheoremThe following two limits hold:sin θlim= 1θ→0θcos θ − 1lim= 0θ→0θProof of the Sine LimitProof.Noticesin θ ≤ θ ≤ tan θDivide by sin θ:θ11 ≤≤sin θcos θTake reciprocals:sin θ θ tan θθsin θ1 ≥≥ cos θ−1cos θ1θAs θ → 0, the left and rightsides tend to 1. So, then,must the middleexpression.Now1 − cos θ1 − cos θ1 + cos θ1 − cos2 θ=·=θθ1 + cos θθ(1 + cos θ)sin2 θsin θcos θ==·θ(1 + cos θ)θ1 + cos θSo1 − cos θsin θcos θlim=lim·limθ→0θθ→0θθ→0 1 + cos θ= 1 · 0 = 0.dsin(x + h) − sin xsin x = limdxh→0h(sin x cos h + cos x sin h) − sin x= limh→0hcos h − 1sin h= sin x · lim+ cos x · limh→0hh→0h= sin x · 0 + cos x · 1 = cos xDerivatives of Sine and CosineTheoremd sin x = cos x.dxProof.From the deﬁnition:(sin x cos h + cos x sin h) − sin x= limh→0hcos h − 1sin h= sin x · lim+ cos x · limh→0hh→0h= sin x · 0 + cos x · 1 = cos xDerivatives of Sine and CosineTheoremd sin x = cos x.dxProof.From the deﬁnition:dsin(x + h) − sin xsin x = limdxh→0hcos h − 1sin h= sin x · lim+ cos x · limh→0hh→0h= sin x · 0 + cos x · 1 = cos xDerivatives of Sine and CosineTheoremd sin x = cos x.dxProof.From the deﬁnition:dsin(x + h) − sin xsin x = limdxh→0h(sin x cos h + cos x sin h) − sin x= limh→0h= sin x · 0 + cos x · 1 = cos xDerivatives of Sine and CosineTheoremd sin x = cos x.dxProof.From the deﬁnition:dsin(x + h) − sin xsin x = limdxh→0h(sin x cos h + cos x sin h) − sin x= limh→0hcos h − 1sin h= sin x · lim+ cos x · limh→0hh→0hDerivatives of Sine and CosineTheoremd sin x = cos x.dxProof.From the deﬁnition:dsin(x + h) − sin xsin x = limdxh→0h(sin x cos h + cos x sin h) − sin x= limh→0hcos h − 1sin h= sin x · lim+ cos x · limh→0hh→0h= sin x · 0 + cos x · 1 = cos xcos xIllustration of Sine and Cosineyxπππ−0π22sin xDocument Outline
• Announcements
• Two important trigonometric limits
• Derivatives of sine and cosine
• Derivatives of tangent and secant

Lesson 10: Derivatives of Trigonometric Functions

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