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Section 2.4 Trig Functions

Let's review the unit circle:

Figure 2.4.1. Unit Circle

A point on the circle has coordinates \((x,y) = (\cos(\theta), \sin(\theta))\text{.}\)

  • \(\displaystyle \left( \dfrac{\sqrt{4}}{2}, \dfrac{\sqrt{0}}{2} \right) = 2 \pi \text{ or } 0 \)

  • \(\displaystyle \left( \dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{1}}{2} \right) = \dfrac{\pi}{6} \)

  • \(\displaystyle \left( \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2} \right) = \dfrac{\pi}{4} \)

  • \(\displaystyle \left( \dfrac{\sqrt{1}}{2}, \dfrac{\sqrt{3}}{2} \right) = \dfrac{\pi}{3} \)

  • \(\displaystyle \left( \dfrac{\sqrt{0}}{2}, \dfrac{\sqrt{4}}{2} \right) = \pi \)

To calculate derivatives, all we know is the limit definition. Let's calculate the derivative of \(\cos(x)\text{:}\)

\begin{align*} \dfrac{d}{dx}(\cos(x)) \amp = \lim_{h \to 0} \dfrac{\cos(x + h) - \cos(x)}{h}\\ \amp = \lim_{h \to 0} \dfrac{\cos(x) \cdot \cos(h) - \sin(x) \cdot \sin(h) - \cos(x)}{h}\\ \amp = \lim_{h \to 0} - \cos(x) \left( \dfrac{1 - \cos(h)}{h} \right) - \sin(x) \left( \dfrac{1 - \sin(h)}{h} \right)\\ \amp = - \cos(x) \left( \lim_{h \to 0} \dfrac{1 - \cos(h)}{h} \right) - \sin(x) \left( \lim_{h \to 0} \dfrac{1 - \sin(h)}{h} \right)\\ \amp = - \cos(x) (0) - \sin(x) (1)\\ \amp = - \sin(x) \end{align*}

Similarly, \(\dfrac{d}{dx}(\sin(x)) = \cos(x)\)

Let's try finding the derivative of \(\tan(x)\) using the quotient rule:

\begin{align*} \dfrac{d}{dx}(\tan(x)) \amp = \dfrac{d}{dx}\left( \dfrac{\sin(x)}{\cos(x)} \right)\\ \amp = \dfrac{\cos(x) \cdot \frac{d}{dx}(\sin(x)) - \sin(x) \cdot \frac{d}{dx}(\cos(x))}{(\cos(x))^2}\\ \amp = \dfrac{\cos(x) \cdot \cos(x) - \sin(x)(- \sin(x))}{(\cos(x))^2}\\ \amp = \dfrac{\cos^2(x) + \sin^2(x)}{\cos^2(x)}\\ \amp = \dfrac{1}{\cos^2(x)}\\ \amp = \sec^2(x) \end{align*}

Similarly, \(\dfrac{d}{dx}(\cot(x)) = - \csc^2(x)\)

So far:

Table 2.4.2. Trig Function Derivatives
\(\dfrac{d}{dx}(\sin(x)) = \cos(x)\)
\(\dfrac{d}{dx}(\cos(x)) = -\sin(x)\)
\(\dfrac{d}{dx}(\tan(x)) = \sec^2(x)\)
\(\dfrac{d}{dx}(\cot(x)) = -\csc^2(x)\)
\(\dfrac{d}{dx}(\sec(x)) = \sec(x) \cdot \tan(x)\)
\(\dfrac{d}{dx}(\csc(x)) = -\csc(x) \cdot \cot(x) \)
Note 2.4.3.

Derivatives of the "co-" funtions are all negative in sign.

Figure 2.4.4. Slopes
The value of \(\cos(x)\) describe the slopes of \(\sin(x)\text{.}\)

Similarly, we could consider the graph of \(\cos(x)\text{;}\) values of \(-\sin(x)\) describe slopes of \(\cos(x)\text{.}\)