In Exercises 61–66, use the method of adding y-coordinates to gra...

Question

In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π.

y = cos x + cos 2x

Accepted Answer

Hello, everyone. We are asked to graph the function Y equals the cosine of X plus three cosine of two X for the interval where X is between or equal to zero and two pi by performing the addition of the Y coordinates, we are given a coordinate plane. Our X axis is measured in pi divided by four intervals. And our Y axis is measured in one unit intervals. First recall that if we are doing the addition of Y coordinates, this means Y will equal the Y value from the first section. So we'll call that Y sub one and that'll be cosine of X will now be Y one and the Y value of the second section. So Y sub two. So three cosine of two, X is now Y sub two. So we are gonna find the Y value for each section and then add them together to find the total Y value. I'm going to make a table. And the first thing in this table I'm going to put are all my X values. So every X value that is on here, I'm going to put down as long as it falls in our interval. So we'll start with zero and pi divided by four pi divided by two, three pi divided by four. Hi five pi divided by four, three pi divided by two, seven pi divided by four and then two pie. This might get a little squished. I'm gonna try to keep it spaced out nicely. So to find why one, we said Y sub one is the cosine of X. So I'm gonna go down my X values and find the cosine of every single one of them. So using a unit circle, the cosine of zero is one, the cosine of pi divided by four would be radical two divided by two which since we are working in a unit scale for the Y axis, I'm gonna make into a decimal of 0.07 pi divided by two on our unit circle is zero three. Pi divided by four is negative radical two divided by two. So that'd be negative 0.7. Pi is negative one five pi divided by four is negative radical two divided by two. So excuse me, negative 0.7 three pi divided by two is zero seven pi divided by four is radical two divided by two. So 0.7 and two pie is one. So we have our Y ones done next, we need to do Y sub two. So that's three multiplied by the cosine of two X. So there's a couple of steps that will happen here. So let's, we'll think through them first, we're gonna take our X value and double it. So zero, double is zero, then we know the cosine of zero is one, three multiplied by one is three for the next one pi divided by four. When we double that, we'd have pi divided by two coin. A pi divided by two is 00, multiplied by three is zero. Pi divided by two. When we double that X will be like pi, the cosine of pi is negative one, negative one multiplied by three is negative three, three pi divided by four. When we multiply it by two is three pi divided by two which the cosine of that is 00, multiplied by three is zero pi as our X. So multiplied by two would be two pi cosine of two pie is one, one multiplied by three is three, five pi divided by four. When we double that we have five pi divided by two. And then that will give us a zero as our cosine of five pi divided by two is zero. At which 0.3 multiplied by zero is zero three pi divided by two. When we multiply it by two, we have three pi, the cosine of three pi would be negative one negative one multiplied by three is negative three. Next we have seven pi divided by four. When we double that, it would be seven pi divided by two. And then the cosine of seven pi divided by two is zero zero, multiplied by three is zero. You might see a pattern here and then two pie. When we double it is four pie, the cosine of four pi would be 11, multiplied by three is three. So we now have our values for Y one and Y two. So we're gonna add those together in this next column. And going straight down one plus three is four zero plus 0.7 is 0.70 plus negative three is negative three. Negative 0.7 plus zero is negative 0.7. Negative one plus three is two, negative 0.7 plus zero is negative 0. zero plus negative three is negative three zero plus 0.7 is 0. and one plus three is four. If it helps you to keep organized to write your new coordinates in a last column here, you absolutely can. You would have 04 pi divided by four and 0.7 pi divided by two and negative three. Three pi divided by four and negative 0. pi and two five pi divided by four and negative 0. three pi divided by two and negative three seven pi divided by four and 0. and then two pie and four. So now that we have all of our points, let's get them on our graph. So I'm gonna stick with the red pen. So that way you can see the spots a little easier. So 04 on our y axis pi divided by four. So the first line in the positive side of the X axis and 0.7. So not all the way up to one but close. And then we have pi divided by two and negative three and then three pi divided by four and negative 0.7 pie and two five pi divided by four, a negative 0.7 three, pi divided by two and negative three, seven pi divided by four and positive 0. and then two pie and four. I'm gonna do my best to connect those so that we still have a little bit of a loop to them. So they're not hard turns. And when I do, we have a W shape and this is what our graph of Y equals the cosine of X plus three, cosine of two X should look like. Have a nice day.

In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = cos x + cos 2x

Key Concepts

Trigonometric Functions

Graphing Techniques

Periodicity and Amplitude

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