3.5.1 Derivatives of Trigonometric Functions

Fundamental Trigonometric Limits

• Two "gateway" limits are used to build all derivative formulas for trigonometric functions.
  • \displaystyle \lim_{x \to 0} \frac{\sin x}{x}=1
  • \displaystyle \lim_{x \to 0} \frac{\cos x-1}{x}=0
• Direct substitution fails (denominator \to 0), but each result can be justified by:
  • Numerical tables
  • Graphs of \sin x and \cos x zoomed near the origin
  • Geometry of the unit circle (arclength vs. chord)

Using the Gateway Limits to Evaluate Other Limits

• Strategy: Rewrite a target limit so that the pattern \sin( ext{something})/(\text{same something}) or (\cos(\text{something})-1)/(\text{same something}) appears.
• Example 1: \displaystyle \lim_{x \to 0} \frac{\sin(4x)}{4}
  • Multiply numerator & denominator by 4 so that the argument and denominator match.
  • \frac{\sin(4x)}{4}=\frac{4\sin(4x)}{4\cdot4x}=\frac{\sin(4x)}{4x}
  • Substitute t=4x (so t\to0)
  • 4\cdot \lim_{t\to0}\frac{\sin t}{t}=4\cdot1=4
• Example 2: \displaystyle \lim_{x \to 0} \frac{\sin(3x)}{\sin(5x)}
  • Divide numerator & denominator by x:
  • \frac{\sin(3x)}{\sin(5x)}=\frac{ \big(\sin(3x)/(3x)\big)\,3}{ \big(\sin(5x)/(5x)\big)\,5}
  • Each bracketed ratio \to1, so the whole limit =3/5.

Derivatives of the Basic Trigonometric Functions

• Using the definition of the derivative together with the two gateway limits, one proves (details in textbook):
  • \displaystyle \frac{d}{dx}[\sin x]=\cos x
  • \displaystyle \frac{d}{dx}[\cos x]=-\sin x
• Geometric intuition:
  • \sin x and \cos x are periodic ➔ their slopes (derivatives) should also oscillate.

Differentiation Examples

Useful rules: Product rule \big(uv\big)'=u'v+uv'; Quotient rule \big(\tfrac{u}{v}\big)'=\dfrac{u'v-uv'}{v^{2}} ; Sum/Difference rule [u\pm v]'=u'\pm v'.

(a) f(x)=e^{x}\cos x

• Product rule
  • f'=\big(e^{x}\big)'\cos x+e^{x}(\cos x)'
  • =e^{x}\cos x+e^{x}(-\sin x)
  • Factor common term: f'(x)=e^{x}\big(\cos x-\sin x\big)

(b) g(x)=\sin x - x\cos x

• Treat as a difference; second term needs product rule.
  1. Derivative of \sin x is \cos x.
  2. For x\cos x: 1\cdot\cos x + x(-\sin x).
• Assemble:
  • g'=\cos x-\big(\cos x - x\sin x\big)=x\sin x

(c) h(x)=\dfrac{1+\sin x}{1-\sin x}

• Quotient rule with u=1+\sin x,\;u'=\cos x and v=1-\sin x,\;v'=-\cos x.
  • h'=\dfrac{u'v-uv'}{v^{2}}=\dfrac{\cos x(1-\sin x)-(1+\sin x)(-\cos x)}{(1-\sin x)^{2}}
  • Expand the numerator:
  • First term: \cos x - \cos x\sin x
  • Second term: +(1+\sin x)\cos x
  • Cancellation: -\cos x\sin x + \cos x\sin x =0
  • Remaining: 2\cos x
  • Result: h'(x)=\dfrac{2\cos x}{\big(1-\sin x\big)^{2}}

Connections & Take-Aways

• Mastery of the two fundamental limits is essential; they recur in almost every proof involving trigonometric derivatives.
• Because all other trig functions (\tan,\cot,\sec,\csc) can be written with \sin and \cos, their derivatives ultimately rely on these basic formulas.
• Algebraic tactics (multiplying/dividing by matching factors, clever substitutions) are often the fastest path to limit evaluation and simplification.
• Periodicity carries over: derivatives of periodic functions generally remain periodic.
• Practical implication: Trigonometric derivatives appear in physics (harmonic motion), engineering (signal processing), and any subject dealing with waves.