Wize High School Grade 12 Pre-Calculus Textbook > Rates of Change

Rates of Change of Trigonometric Functions

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Rates of Change of Trigonometric Functions
The rate of change of a trigonometric function can be solved used both algebraic and graphical methods. 

Example

The following graph is a sketch of the function y=2cos⁡(θ−π4)+1y=2\cos\Big(\theta-\dfrac{\pi}{4}\Big)+1y=2cos(θ−4π​)+1


The average rate of change between the interval π4≤θ≤5π4\dfrac{\pi}{4}\leq{}\theta\leq{}\dfrac{5\pi}{4}4π​≤θ≤45π​ is:


f(x2)−f(x1)x2−x1=(2cos⁡(5π4−π4)+1)−(2cos⁡(π4−π4)+1)5π4−π4=(2cos⁡(π)+1)−(2cos⁡(0)+1)π=−1−(3)π=−4π\begin{array}{rcl}
\dfrac{f(x_2)-f(x_1)}{x_2-x_1}&=&\dfrac{\Bigg(2\cos\Big(\dfrac{5\pi}{4}-\dfrac{\pi}{4}\Big)+1\Bigg)-\Bigg(2\cos\Big(\dfrac{\pi}{4}-\dfrac{\pi}{4}\Big)+1\Bigg)}{\dfrac{5\pi}{4}-\dfrac{\pi}{4}}\\\\
&=&\dfrac{(2\cos(\pi)+1)-(2\cos(0)+1)}{\pi}\\\\
&=&\dfrac{-1-(3)}{\pi}\\\\
&=&\dfrac{-4}{\pi}
\end{array}x2​−x1​f(x2​)−f(x1​)​​====​45π​−4π​(2cos(45π​−4π​)+1)−(2cos(4π​−4π​)+1)​π(2cos(π)+1)−(2cos(0)+1)​π−1−(3)​π−4​​

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Example: Rates of Change of Trigonometric Functions
Determine the rate of change of the trigonometric function f(θ)=2cot⁡(2θ)+1f(\theta)=2\cot(2\theta)+1f(θ)=2cot(2θ)+1 on the interval π8≤θ≤π4\dfrac{\pi}{8}\leq{}\theta\leq{}\dfrac{\pi}{4}8π​≤θ≤4π​.

msec=(2cot⁡(2⋅π4)+1)−(2cot⁡(2⋅π8)+1)π4−π8=2(cot⁡(π2)−cot⁡(π4))π8=16(0−1)π=−16π\begin{array}{rcl}
m_{sec}&=&\dfrac{\Bigg(2\cot\Big(2\cdot\dfrac{\pi}{4}\Big)+1\Bigg)-\Bigg(2\cot\Big(2\cdot\dfrac{\pi}{8}\Big)+1\Bigg)}{\dfrac{\pi}{4}-\dfrac{\pi}{8}}\\\\
&=&\dfrac{2\Bigg(\cot\Big(\dfrac{\pi}{2}\Big)-\cot\Big(\dfrac{\pi}{4}\Big)\Bigg)}{\dfrac{\pi}{8}}\\\\
&=&\dfrac{16(0-1)}{\pi}\\\\
&=&-\dfrac{16}{\pi}
\end{array}msec​​====​4π​−8π​(2cot(2⋅4π​)+1)−(2cot(2⋅8π​)+1)​8π​2(cot(2π​)−cot(4π​))​π16(0−1)​−π16​​

Practice: Rates of Change of Trigonometric Functions
Determine the average rate of change of the function f(θ)=2cos⁡(3θ−π)−2f(\theta)=2\cos(3\theta-\pi)-2f(θ)=2cos(3θ−π)−2 on the interval −3π4≤θ≤5π6-\dfrac{3\pi}{4}\leq{}\theta\leq{}\dfrac{5\pi}{6}−43π​≤θ≤65π​.

A) 2419π2\dfrac{24}{19\pi\sqrt{2}}19π2​24​ 

B) −2419π2-\dfrac{24}{19\pi\sqrt{2}}−19π2​24​ 

C) 24(1+2)19π2\dfrac{24(1+\sqrt{2})}{19\pi\sqrt{2}}19π2​24(1+2​)​ 

D) 24(−1+2)19π2\dfrac{24(-1+\sqrt{2})}{19\pi\sqrt{2}}19π2​24(−1+2​)​

I don't know

Practice: Rates of Change of Trigonometric Functions
Alex, Rosa, and Diego were out jet-skiing when they noticed the weather started to turn. They turned their jet-skis around and docked at their harbor. The jet-skis will rise and fall with the waves of the water and d(t)=2cos⁡(3π4t)d(t)=2\cos\Big(\dfrac{3\pi{}}{4}t\Big)d(t)=2cos(43π​t) represents the displacement of the jet-skis from the dock in meters. 

a. Determine the average rate of change of the displacement of the jet-skis over the first

10

minutes.

b. Determine the instantaneous rate of change when

t=5

minutes. Let

h=0.001

. Round to the nearest thousandth.

I don't know

Practice: Rates of Change of Trigonometric Functions
Let y=tan⁡θy=\tan{\theta}y=tanθ.

Practice: Rates of Change of Trigonometric Functions
Estimate the instantaneous rate of change of the function y=tan⁡θy=\tan{\theta}y=tanθ  for the following values of θ.\theta. θ. Use h=0.001h=0.001h=0.001. 

\theta=-\pi