Calculus Small Changes, Big Ideas

Section 4.3 Trigonometric functions

• Simple harmonic motion:

  \begin{equation*} y''(t)=-ky(t) \end{equation*}

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• Can’t use slope fields, but can use a modified Euler’s method to reconstruct the curve given initial conditions on \(y\) and \(y'\text{.}\)

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• Define \(\sin x\) as the solution to \(y''=-y\) with \(y(0)=0\) and \(y'(0)=1\text{,}\) and define \(\cos x\) as the solution to the same differential equation with \(y(0)=1\) and \(y'(0)=0\text{.}\)

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• Can show that \(\DS\dv{x} \sin x=\cos x\) by letting \(C(x)=\DS\dv{x} \sin x\) and noticing that \(C\) also satisfies the same differential equation and has the same initial conditions as \(\cos x\text{.}\) Similarly, can show that \(\DS\dv{x} \cos x=-\sin x\text{.}\)

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• Other properties such as the sum/difference identities and the fact that \(\sin^2 x + \cos^2 x = 1\) can also be shown by using the differential equation and initial conditions.

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• Quotient Rule on \(\sin x\) and \(\cos x\) gives the derivatives of the other trig functions:

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  \begin{align*} \dv{x}\tan x \amp = \sec^{2} x \amp \dv{x}\sec x \amp = \sec x \tan x\\ \dv{x}\cot x \amp = -\csc^{2} x \amp \dv{x}\csc x \amp = -\csc x \cot x \end{align*}
  Every one of these can be thought of backwards to become an antiderivative formula.

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• Briefly consider damped harmonic motion:

  \begin{equation*} \ee^{-at}\cos bt \end{equation*}

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