Section 3.5 Derivatives of Trigonometric Functions
Recall: \(\sin(A+B)=\sin A \cos B + \cos A \sin B\)
Recall: \(\displaystyle \sin^2(\theta/2) = \frac{1-\cos\theta}{2}\)
Theorem 3.5.1. Derivative of sine.
\(\displaystyle\frac{d}{dx} \sin x = \cos x\)
Proof.
\begin{align*}
\lim_{h\rightarrow 0} \frac{\sin(x+h)-\sin(x)}{h} \amp = \lim_{h\rightarrow 0} \frac{\sin(x)\cos(h)+\cos(x)\sin(h)-\sin(x)}{h}\\
\amp = \lim_{h\rightarrow 0} \frac{\sin(x)(\cos(h)-1)+\cos(x)\sin(h)}{h}\\
\amp = \lim_{h\rightarrow 0} \frac{\sin(x)(\cos(h)-1)}{h} + \lim_{h\rightarrow 0}\frac{\cos(x)\sin(h)}{h}\\
\amp = \sin(x) \lim_{h\rightarrow 0} \frac{\cos(h)-1}{h} + \cos(x)\lim_{h\rightarrow 0}\frac{\sin(h)}{h}\\
\amp = \sin(x) \lim_{h\rightarrow 0} \frac{\cos(h)-1}{h} + \cos(x)\lim_{h\rightarrow 0}\frac{\sin(h)}{h}\\
\amp = \sin(x) \lim_{h\rightarrow 0} \frac{-2\sin^2\left(\frac{h}{2}\right)}{h} + \cos(x)\lim_{h\rightarrow 0}\frac{\sin(h)}{h}\\
\amp = \sin(x) \lim_{h\rightarrow 0} -\sin\left(\frac{h}{2}\right)\frac{\sin\left(\frac{h}{2}\right)}{h/2} + \cos(x)\lim_{h\rightarrow 0}\frac{\sin(h)}{h}\\
\amp = \sin(x) \cdot 0 \cdot 1 + \cos(x)\cdot 1\\
\amp = \cos(x)
\end{align*}
Fact 3.5.2.
\(\displaystyle\frac{d}{dx} \cos x = -\sin x\)
Example 3.5.3.
Find the following:
\(\displaystyle \displaystyle \frac{d}{dx} \frac{7}{\sqrt{x}}+x\sin x\)
\(\displaystyle \displaystyle \frac{d}{d\theta} \frac{\sin \theta}{1+\cos \theta}\)
\(\displaystyle \displaystyle \frac{d}{dx} x^8 \sin x \cos x\)
\(\displaystyle \displaystyle \frac{d}{dx} 3 \sec x + 5 \tan x\)
Fact 3.5.4. Derivatives of trig functions.
\begin{align*}
\frac{d}{d\theta} \sin\theta \amp = \cos\theta\\
\frac{d}{d\theta} \cos\theta \amp = -\sin\theta\\
\frac{d}{d\theta} \tan\theta \amp = \sec^2\theta\\
\frac{d}{d\theta} \csc\theta \amp = -\csc\theta \cot\theta\\
\frac{d}{d\theta} \sec\theta \amp = \sec\theta\tan\theta\\
\frac{d}{d\theta} \cot\theta \amp = -\csc^2\theta
\end{align*}
Example 3.5.5. Harmonic motion.
A weight hanging from a spring is stretched 7 inches beyond its resting point and is released. Its position at any time \(t\) is given by \(s=7\cos t\) (measured in inches, with positive up).
What is the displacement when \(t=0\text{?}\) when \(t=\frac{\pi}{3}\text{?}\)
Find formulas for the velocity, acceleration, and jerk.
What is the velocity when \(t=\frac{\pi}{3}\text{?}\)
Answer.
\(7\text{;}\) \(\frac{7}{2}\)
velocity: \(-7\sin t\text{,}\) acceleration: \(-7\cos t\text{,}\) jerk: \(7\sin t\)
\(\displaystyle \frac{-7\sqrt{3}}{2}\)
