[Technology Exercise]

a. Graph y = cos x and ...

Question

[Technology Exercise]

a. Graph y = cos x and y = sec x together for −3π/2 ≤ x ≤ 3π/2. Comment on the behavior of sec x in relation to the signs and values of cos x.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Graph the functions Y equals sin of X and Y equals cosicant of X on the same coordinate plane and consider the domain negative pi is less than or equal to X and X is less than or equal to pi. Describe the relationship between the two graphs. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. Our first answers we're asked to graph both functions for Y equals sin of X and Y equals cosican of Xx on the same coordinate planes, so we're plotting them on the same graph together. That's our first answer we're trying to solve for. Our second answer is we're trying to describe the relationship between these two graphs, and that is our second and final answer that we're ultimately trying to solve for. So now that we know what we're ultimately trying to solve for, let us first recall and note that we need, uh, first off, in order to keep things simple, as we should recall, We need to graph one graph first, so let's start with graphing one of the graphs first. So let's start with Y equals sin of X, and then we'll graph Y equals cosicant of X. So focusing our attention on Y equals sin of x first, let us note that Y equals sin of X, focusing on this. Particular function first, let us know that the period P is equal to 2 pi for this particular prong, and let us also note once again that the domain is negative pi is less than or equal to X and X is less than or equal to positive pi. Fantastic. So the domain is equivalent to 1 since i minus negative pi is equal to 2 pi. So now we need to divide the period into 4 equal parts, so we need to take our period and divide it by 4, so we need to take 2 pi divided by 4, which will give us an answer of pi divided by 2. So now we need to determine the X values. So let us start with X1, and X1 is going to be simply equal to negative high. And then we need to solve for X2, X3. X4 and X 5. Fantastic. So X2 is going to be equal to X1. Plus, I divided by 2, so plugging in our value of X1 into our expression, it will give us an answer of when we plug in our values and we solve or plug it into our calculator and solve, we'll get negative pi divided by 2, and then X3 is going to be equal to X2. So we take our value of X2 and we add pi divided by 2, and when we take X2 and we add. Pi divided by 2, we will find that it's equal to 0. And for X4, we need to take our value for X3, and we need to add pi divided by 2, and this is going to be equal to divided by 2, whereas X2 is equal to negative pi divided by 2. And finally, X5 is going to be equal to the value of X4 plus pi divided by 2, which is going to be equal to simply pi. Fantastic. So now our next step. we need to evaluate the function at the specific X value, so we need to find that Y1 is going to be equal to Y of X1, since we're putting in a value of X1 in for X. So this will mean that it's gonna be Y. So it's gonna be equal to Y of negative pi, which is gonna be equal to sine of negative pi, which is gonna simply give us 0. Fantastic. So, following this similar methodology, We need to note that We are going to get the following for the rest, so we're going to be following this pattern, we're going to get the following results for Y2, Y3, Y4, and Y5 because once again we're we're trying to evaluate the functions at each of our X values that we solve for. So instead of rewriting our expression out every single time, I'm just going to give you the different values for Y2. Y3 and Y4 knowing once again that Y2 is going to be equal to Y of X2, which is going to be simply equal to -1. Y3 is going to be X of X. So it's going to be Y of X3, which is going to be simply equal to 0. And once again you're just plugging in our value of X2 X3 into our expression for X. And for Y4, it's going to be it's going to be equal to Y of X4, which is going to give us an answer of 1, whereas Y2 is -1, and then Y5 is going to be equal to Y of X5, which is going to give us an answer of 0. So now what we need to do is we need to plot. The following points negative, 0. And we also need to plot negative pi divided by 2.1, and we need to plot 0.0, which, as we should recall a note, that's the origin point. We also need to plot positive pi divided by 2.1, and we need to plot in parentheses. And when we plot these four coordinate points, we also need to connect them with a smooth curve. So looking at our graph, which we have a blank graph here of our coordinate planes, so let's plot our different points. So, I'm gonna plot the different points in blue. So we have our point at negative pi 0 is our first corner point, and then we have our point at negative pi divided by 2, -1. Then we have our corner point at the origin point, and then we have one at positive pi divided by 2.1, and then we have our final point at 0. So when we connect them with a smooth curve, we should get the following graph. Fantastic. So now, our next step that we need to take. we need to graph Y equals cosicant of X by drawing the vertical asymptotes first. So let us recall and note that the vertical asymptotes on the integral multiples of i should pass through the X intercepts of Y equals sin of X. So that will mean that our vertical asymptotes will have to be at negative pi. And at positive pi, which I'm using red dotted lines to denote our vertical asymptotes. So now we need to sketch the graph of Y equals cosicant of X using the vertical asymptotes as our guide. And as we should recall a note that since sine and cosine are reciprocal functions, the curve for cosicant should touch the curve at, and let's make a note of this in. And well, let's make a note of it in red. So let us note that when we make a note of this in red, that the curve, so once again, we need to know that sign, that sends sign and cosine are reciprocal functions. The curve for cosicant should touch the curve of sign at X equals negative pi divided by 2, and at X equals divided by 2. And where the Y values are gonna be Y is equal to plus or minus 1, respectfully. So this is because the reciprocal of -1 and 1 are -1 and 1 as well. So when we do that, when we graph these graphs, we should produce the following graph. So I've gone ahead and plugged in both of our functions into some graphing software, and as you can see, we should get the following graph. And really quick here I'm gonna Erase what we drew so we could see our graph graph properly. So when we plot our second graph for Y equals cosican of X, we should produce the following graph. And once again, I plug this into my graphing software and I produce the following graph. And as you can see, Our Y equals cos of X is represented in red, and our curve for Y equals sin of X is represented in blue. And as you can see, it's within our vertical asymptotes, and let us note that the curves follow the vertical asymptote, but they do not cross or go beyond the vertical asymptotes. And as you could see, It could be observed from the graphs that the cosicant function has vertical asymptotes and is undefined at these particular points where the sine function is 0, where X is equal to negative pi, X is equal to 0, and X is equal to positive pi, and the cosicant function also touches the points where the sine function has peaks at of X is equal to 1, and sine of X is equal to -1. And this is because sine and cosine are reciprocal functions, and the reciprocal at these points are also 1 and -1. So, with that said, quickly looking at our graph once again, so our Y equals cosican of X resembles a parabolic shape, and it touches the peak of our curve in the positive quadrant for our Y equals sin of X. Then we have a parabola that's inverted that also touches the bottom peak, the peak that's pointing towards the bottom or rather more like a trough. And they both touch the mountain peak, so it's gonna be an inverse parabolic shape, and that's it. That is our final answers. Hooray, we did it. So to quickly recap. Our final answer once again for the second part, which is describe the relationship between the two graphs. The cosicant function has vertical asymptotes at the point where the sine function is zero. The cosicant function also touches the points where the sine function has peaks, as we discussed, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

[Technology Exercise]

a. Graph y = cos x and y = sec x together for −3π/2 ≤ x ≤ 3π/2. Comment on the behavior of sec x in relation to the signs and values of cos x.

Key Concepts

Cosine Function

Secant Function

Graphing Trigonometric Functions

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