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

Question

In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = 3 cos x + sin x

Accepted Answer

Hello, everyone. We are asked to graph the function Y equals eight cosine of X plus three sin of X when X is between or equal to zero and two pi we're gonna do this by performing the edition of Y coordinates. We are given a coordinate plane with our X axis labeled in pi divided by four intervals. And our Y axis labeled in one unit recall that the addition of the Y coordinates says that why will equal the Y value of the first term plus the Y value of the second term? So I'm gonna call those Y one and Y two. So in our example, Y one will be eight cosine of X Y two will be three sin of X. I'm gonna start by making a large table because we have a lot of X values to look at here. So we are gonna use all the intervals that are on our coordinate plane that fall between or equal to zero and two pi. So that's zero pi divided by four high, divided by two, the three pi divided by four pie, five pi divided by four, the three pi divided by two, seven pi divided by four and two pi and since Y one is looking for the cosine of X, I wanna make a column that is the values of the cosine of X, this will just make it easier to do the multiplication down the road. So the cosine of zero is one and I'm using a unit circle for this. So feel free to get through this quicker if you want to, the cosine of pi divided by four is radical two divided by two. Since we are graphing this, I'm gonna use this as a decimal. So that's gonna be 0.7. The cosine of pi divided by two is zero. The cosine of three pi divided by four is negative radical two divided by two. So negative 0.7 the cosine of pi is negative one. The cosine of five pi divided by four is negative radical two divided by two. So negative 0.7 the cosine of three pi divided by two is zero. The cosine of seven pi divided by four is radical two divided by two. So 0.7 and the cosine of two pie is one. So now I'm gonna find our Y one which is eight multiplied by the cosine of X. So I'm taking every value from my cosine column and multiplying it by eight. So we have 85. negative 5.6 negative eight, negative 5. 0.68 because I took each value from the cosine column and multiplied it by eight. Next, I'm gonna find the sign of each X value to make it easier to multiply for my Y two. So the sign of zero is zero. The sign of pi divided by four is radical two divided by two. So 0.7 the sign of pi divided by two is one. The sign of three pi divided by four is radical two divided by two. So 0.7 the sign of pi is zero. The sign of five pi divided by four is negative radical two divided by two. So negative 0.7 the sign of three pi divided by two is negative one. The sign of seven pi divided by four is negative radical two divided by two. So negative 0.7 and the sign of two pie is zero. So now this will make it a lot easier to find Y two which is three multiplied by the sin of X. So we're gonna take every volume of that sign column and multiply it by three. So we have zero 2.1 three, 2.10 negative 2. negative three, negative 2.10. All right. So we're getting there, we now get to find our Y value. So we're gonna add Y one plus Y two. So make sure you know which columns those are. So that on my screen it's the 2nd and 4th column or I'm sorry, it's the 3rd and 5th column. I forgot to count my X's. So I'm going to draw a little arrow just so we know which ones I'm looking at. I'm gonna add those directly across. So eight plus zero is eight 5.6 plus 2.1 is 7.7 zero plus three is three negative 5.6 plus 2.1 is negative 3.5. Uh negative eight plus zero is negative eight, negative 5.6 plus negative 2.1 is negative 7. zero plus negative three is negative three 5. plus negative 2.1 is 3.5 and eight plus zero is eight. If it helps you to do so to write out each of our coordinates here, feel free to make another column where we have X and Y. I'm gonna list them down quickly just to help us with graphing. So we'll have 08 pi over 47.7 pi divided by two and three three pi divided by four and negative 3. pi and negative eight five pi divided by four and negative 7. three pi divided by two and negative three seven pi divided by four and 3. and two pie and eight. All right. Now let's get those points on our graph. So 080 on the X eight on the Y should be on our Y axis pi divided by four and 7.7. So it's a little lower than the eighth but not all the way down to the seven pi divided by two and three, three pi divided by four and negative 3.5. So a little bit between negative three and negative four high and negative eight five pi divided by four and negative 7.7. So between negative seven and negative eight three pi divided by two and negative three seven pi divided by four and positive 3.5. So between three and four and two pie and eight, I'm gonna do my best to connect these points with a wave, not a sharp turn, but based on the points we plotted and the values we found this should be what our graph would 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 = 3 cos x + sin x

Key Concepts

Graphing Trigonometric Functions

Superposition of Functions (Adding y-coordinates)

Amplitude and Phase Shift in Combined Trigonometric Functions

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