Preview Activity 10.1.1.
After graduating, you decide to start a self-driving car company because it didn’t look too hard based on a few articles you scrolled past online. You decide to call your company Steer Clear and start working on the self-driving part of the car. Your first task is to understand how the location tracking equipment you bought online works. There are two parts to your location tracking system: a receiver box and a tracker that you put in the object you want to track.
(a)
After putting the tracker in your pocket, you get in a car and drive around the receiver box. When you download the data about your drive, you notice that the software is outputting a vector from the receiver box to the tracker. The software also seems to love math, because these vectors are written in terms of trig functions.
Below are some of the output vectors from the software. Evaluate each of these vectors and draw these vectors with initial point at the origin on the axes in Figure 10.1.2.
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\(\displaystyle \langle \cos(0), \sin(0) \rangle\)
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\(\displaystyle \langle \cos\left(\frac{\pi}{2}\right), \sin\left(\frac{\pi}{2}\right)\rangle\)
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\(\displaystyle \langle \cos(\pi), \sin(\pi) \rangle \)
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\(\displaystyle \langle \cos\left(\frac{3\pi}{2}\right), \sin\left(\frac{3\pi}{2}\right)\rangle\)
(b)
Below are a few more output vectors from your software. Evaluate each of these vectors and draw each of these vectors with initial point at the origin on the axes in Figure 10.1.2.
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\(\displaystyle \langle \cos\left(\frac{\pi}{4}\right), \sin\left(\frac{\pi}{4}\right) \rangle\)
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\(\displaystyle \langle \cos\left(\frac{3\pi}{4}\right), \sin\left(\frac{3\pi}{4}\right) \rangle\)
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\(\displaystyle \langle \cos\left(\frac{5\pi}{4}\right), \sin\left(\frac{5\pi}{4}\right) \rangle\)
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\(\displaystyle \langle \cos\left( \frac{7\pi}{4}\right), \sin\left(\frac{7\pi}{4}\right) \rangle\)
(c)
Based on the data you are seeing from the two sets of vector outputs of your software, it appears that the output of your software is the set of vectors of the form \(\langle \cos(t), \sin(t) \rangle\text{,}\) where \(t\) goes from \(0\) to \(2 \pi\text{.}\) Add to Figure 10.1.2 a sketch the set of terminal points (only) of vectors \(\langle \cos(t), \sin(t) \rangle\) plotted in standard position (with initial point at the origin). You should allow \(t\) to assume values from \(0\) to \(2 \pi\text{.}\) Write a couple sentences about what path your drive took.
(d)
What part of the path you drove is described by \(\langle \cos(t), \sin(t) \rangle\text{,}\) where \(t\) goes from 0 to \(\pi\text{?}\) What would the path be if you drove along \(\langle \cos(t), \sin(t) \rangle\) for \(t\) from 0 to \(4 \pi\text{?}\)
