Lesson video

Hello, I'm Mrs. Lashley, and I'm gonna be talking you through the lesson today.

I really hope you're willing to try your best, and even if it gets challenging, remember I'm here to support you.

So, our lesson outcome today is to be able to read values from the graphs and can identify how many solutions exist within a given range, and this is for the trigonometric function graphs.

On the screen, there are some keywords that you've met before in your learning, and I'll be making a use of during the lesson, so let's have a look at them now together.

So trigonometric functions are commonly defined as ratios of two sides of a right-angled triangle for a given angle.

The sine of an angle is the y-coordinate of point P on the triangle formed inside the unit circle, whereas the cosine of an angle is the x-coordinate of point P on the triangle formed inside the unit circle.

The tangent of an angle is the y-coordinate of point Q on the triangle which extends from the unit circle.

And for a repeating function, the period is the distance of one repetition of the entire function.

The definitions about sine, cosine, and tangent are referring to the diagram on the slide, so you may wish to pause the video so that you refer to the definitions and the diagram before we move on.

So, in this lesson on interpreting the trigonometric graphs, we're gonna break the lesson into two learning cycles.

In the first learning cycle, we're gonna focus on reading from the graphs, and then when we move to the second learning cycle, we're going to look at how many solutions, the number of solutions, within a given range.

So, let's make a start looking at how we read from the graphs of trigonometric functions.

So, with some graphs, it's easy to be accurate when reading off values, and you've met many graphs by this point of your learning.

So, if we look at this graph here, it's a linear graph and it's part of the graph of y equals negative 2x plus 0.

5.

So, you can calculate the gradient is negative 2, and it's got a y-intercept of 0.

5.

So, what is the value of x when y equals 0.

3? Pause the video and try and read that from that graph by yourself before I show you how you should have done it when you press play.

So to get the value of x when y equals 0.

3, we can go across; we know where y is equal to 0.

3, that is a horizontal line, so we can go across to our graph at 0.

3 and then drop down to read off its x coordinate, and in this case it's 0.

1.

So here is a very easy way of pulling off the values.

What about when y equals 0.

15? So, pause the video and try that for yourself before I show you the answer when you press play.

So, 0.

15 is going to be halfway between 0.

1 and 0.

2 using our y-axis, and we're going to go across until it intersects the y equals negative 2x plus a half graph and then read off the x value.

However, this time it's much more difficult to read that x value.

I've written x is 0.

175, and that's because of the scale on the axes.

In some cases, you'll need to estimate.

So, for this linear graph, we do have the algebraic skills to be set that up as an equation.

We could say that y equals 0.

15, therefore, 0.

15 is equal to negative 2x plus a half, and we can solve that to get the exact value of x.

Mathematically, we can't always solve those equations.

Linear equations yes, quadratic equations yes, but as we move up the polynomial degree, then actually get to a point where you cannot solve them algebraically.

So, reading them off is a good way of doing it, but sometimes it has to be an estimate.

So here is a check for you.

Solve cosine of x equals 0.

5 for the interval of 0 to 360 degrees.

So, is it a, 60 degrees, b, 120 degrees, or c, 300 degrees? The graph you're going to use is there, and the graph is y equals cosine of x for the interval of 0 to 360.

So, pause the video and have a think about what we just did with the linear graph.

Press play when you're ready to move on.

So, if this is the graph of y equals cosine x and we're looking for when cosine x is equal to 0.

5, then we're actually asking what values of x give you a y value of 0.

5.

So, if we draw our horizontal line at y equals 0.

5, where does it intersect the cosine graph? Well, it intersects the cosine graph twice, at 60 degrees and 300 degrees.

And so, we've got two solutions to that equation.

So, here's another one.

Solve the equation cosine of x is equal to minus 0.

5 for the interval of 0 to 180.

So, pause the video and have a go at that, and then when you're ready to check, press play.

So, this time we're looking at what values of x have a y value of minus 0.

5.

Well, where is minus 0.

5? Minus 0.

5 is obviously that horizontal line, but what's more important as well is our solution was for the interval of 0 to 180.

So, we didn't want to extend that line through the full graph.

Our graph was plotted between 0 and 360, but we were actually restricting that to 0 to 180.

So, there is only one solution here.

It only intersects the cosine graph at one point, and that's 120 degrees.

So, when we are solving trigonometric equations, it's really important that we are looking at what the domain is.

What interval of x values are we taking? So, here's another one for you to try.

Cosine of x is equal to minus 0.

7 for the interval of 0 to 360.

So, pause the video, and when you're ready to check, press play.

So minus 0.

7 is a horizontal line.

We are going from 0 up to 360 because that is the range of x values that we are being asked for, and it intersects the cosine graph twice.

One at 135 and one at 225.

And we can use the scale there to calculate those values.

There was four squares for every 30 degrees, so it was 7.

5 degrees per smaller square, the increment of the scale.

So, to two decimal places, the values of x when you solve cosine of x is minus 0.

7 are actually 134.

43 degrees and 225.

57 degrees.

So, the graph allowed us to make a pretty good estimate.

So, there is a way of solving that equation and getting a more accurate value, but the reading off the graph has given us a good estimate.

So, we're at the first task for you, which includes three questions.

The first question is on the screen here.

So, we've got the cosine graph once again plotted between 0 and 360 degrees.

Do you remember all of the features? The maximum of 1, the minimum of minus 1.

It's a wave, and it's got a period of 360 degrees.

It repeats itself every 360 degrees, so we can only see one period for this interval.

So, for your questions, question part a is solve cosine of x equals 0.

7 for 0 to 360 degrees.

Part b is solving cosine x equals 0.

15 for the values of x between 0 and 90.

And part c is solve cosine of x equals minus 0.

85 for the values of x between 180 and 360.

So really being careful on which part of the graph you're actually reading from.

Pause the video whilst you do that, and then when you're ready to go up to the next two questions, press play.

So, question two, sketch the graph of y equals sine of x for the values of x between 0 and 360.

Do that on the axes that you've been given.

You can use your calculator to get tabled the values.

You could go on to Desmos or GeoGebra and look it up if you've sort of forgotten some of the features.

And then for question three, you're going to use your graph to find solutions, and there you can see the equations that you're going to solve.

So, pause the video whilst you do that.

When you press play, we're going to go through our answers to questions one, two, and three.

So, question one, the answers are on this screen.

On part b, 82 degrees, it would have been an estimate, so you needed to put your sort of best estimate, and hopefully you've gone for some a value that's close to 82 degrees.

But how should you have been doing it? Well, for part A, it was across the full interval; it was across the full domain.

So, you should have drawn a horizontal line at 0.

7, y equals 0.

7, and then when it intersected the cosine graph, which intersected twice then you would drop perpendicular to that to get the x value, and the x values were representing our theta or our angle.

On b, it was only between 0 and 90 that you were focusing on.

So you need to draw a line between you need to draw the line y equals 0.

15 for the domain of 0 to 90 and where that hits the graph which only hits it at one point drop down from that to read off the value and that's where you'll have your sort of estimate of 82 degrees.

For part C, it was cosine of x equals minus 0.

85, but it was for the interval of 180 to 360.

So, 180 is the halfway point of this period up to the end.

Minus 0.

85 obviously is a negative y value, and so where did it hit the graph? So you needed to read that off and then get your estimate of 210.

Question two: you needed to sketch the graph of y equals sine of x.

So, the key points were that it looks like a wave.

It's symmetrical between 0 and 180, and then 180 and 360.

It's got a local maximum at 91 and a minimum at 270 negative 1.

So, once you've drawn that, you're then going to make use of it for question three.

So, question three, the solutions are on the screen.

So, for the first one, it was what values of x does sine of x equal minus 0.

5, and it was for the full domain from 0 to 360.

So, when we draw our line of minus 0.

5, it does intersect twice and intersects past the 180 degrees; it actually intersects at 210 degrees and 330 degrees.

On part b, it was what values of x does sine of x take 0.

25, and this time it was between 90 and 270, the sort of middle section of the sine curve we can see.

Draw your horizontal line at 0.

25.

Where does it intersect? Well, it actually intersects once because the sine curve goes from the positive y values to the negative y values within that domain, and so we're above the x-axis because our y equals 0.

25, so it's only hitting it once, 165 degrees.

And on part c, what values of x does sine of x equal minus 0.

6, and this was between 180 and 360.

So, when we go across at minus 0.

6 and read off those values, what are they? So, 217 degrees and 323 degrees are rounded to the nearest degree, so again, if you've given an answer in that area, then of course mark yourself correct.

So now we've done learning cycle one, where we know how to read from the graph, we're now going to look at the number of solutions within a given range on a trigonometric graph.

Jun says it's really easy to tell how many solutions there are using the graph.

I just have to pay attention to what values x can take.

Alex says if that's true, then you'll have no trouble answering my questions.

So, Alex is a quiz master, and he's gonna ask Jun some questions.

So, for this check, how many solutions will there be? So, this is the graph of y equals cosine x between 0 degrees and 360 degrees.

So, solve cosine x for minus 0.

5.

We don't actually need to solve it.

I just want to know how many solutions there will be to that equation.

So, pause the video and then when you're ready to check, press play.

Jun said two.

Did you say two? So here is minus 0.

5; here is y equals minus 0.

5.

How many times does it intersect with our graph? Twice.

So, Alex says well done; that's correct.

Hopefully, you got that right as well.

Here is another one, then.

How many solutions will there be for the equation cosine of x equals 0.

35? Pause the video, and then when you're ready to check, press play.

Jun has stated one.

Do you agree with Jun? So, we're looking at the domain of 0 to 270.

So, we're not going across the full graph that we can see here.

We're only going between the 0 up to 270.

And Alex says correct.

So, because of that restricted domain, there is only one solution.

Okay, what about this? Cosine of x is equal to 0.

5 is the equation that we're trying to think about the number of solutions for 0 to 720.

So how many solutions will there be? Pause the video, and then when you're ready to check, press play.

Jun said two.

Did you say two? So, if we draw the line at 0.

5, it intersects the graph at two places.

But Alex says nope.

Okay, so why is Jun wrong? Hopefully, you didn't say two.

So, you should have the answer here.

What should the answer be and explain why? So, pause the video if you did say two.

Don't worry, but now think about why that is not the correct answer.

Press play when you're ready to check.

So why is Jun wrong? Well, because of the interval.

The graph here is only drawn between 0 and 360.

But we were solving the equation over a much larger interval, twice as large.

So, he only gave those solutions for that interval of x values.

So, what should the answer be? Well, the function has a period of 360 degrees, so cosine and sine, but we're focusing on cosine here, has a period of 360 degrees.

Therefore, this shape, this graph, repeats again between 360 degrees and 720 degrees.

So, if there are two solutions in this first period, then there will be two further solutions in the next period.

And so, there will be four in total.

So, the correct answer, Jun, should have said that there are four solutions to this equation.

So, really be mindful of what the interval for your solutions is.

So, the sine and cosine functions both have a period of 360 degrees.

So, this means they repeat every 360.

The tangent function has a period of 180 degrees, and so this means it repeats every 180 degrees.

So, here's another check.

Solve the equation that sine of x equals minus 0.

5 for the interval of 0 to 720.

How many solutions will there be? So, be careful with this.

Pause the video when you're ready to check.

Press play.

Jun says four.

Do you agree with Jun? Alex says correct because we can see in the period of 0 to 360 there are two solutions, and that means when we get to 0 to 720, which will be two periods of sine, there will be four solutions.

So, we're up to the last task of the lesson, which is about how many solutions.

So, how many solutions will there be for cosine x equals 0.

921 between 0 and 360? For part b, how many will there be for cosine x equals 0.

921 between 0 and 720? And for part c, how many solutions will there be for cosine x equals 0.

921 between 0 and 540? Pause the video, have a go at that question, and then when you're ready for the next one, press play.

So, here is question two.

Similar style, but this time we're looking at the sine function.

So, for part a, for part b, and for part c, how many solutions to those equations will there be? Pause the video when you're ready for the final question of task B.

Press play.

So now we're looking at the tangent function.

Same thing, how many solutions will there be? So, pause the video, and when you press play, we're going to go through our answers to one, two, and three.

So, on question one, how many solutions will there be? We've got the graph between 0 and 360 for cosine, and it does have a period of 360 degrees.

So, on question part a, there would be two solutions.

If you go across at 0.

921, you would hit the graph twice.

For part b, it was the same equation, but this time the interval has been doubled.

So, the amount of solutions will also double for this case.

On question part c, it was the same equation, but if we look now at our interval, it goes between 0 and 540.

Well, 540 is 180 more than 360.

So, it's half a period.

It's going to be the section of the graph we can see between 0 and 180 that we're going to see additionally.

So, how many times will the line y equals 0.

921 cross that? Well, it'll cross it twice between 0 and 360, and it crosses it once between 0 and 180.

So, that's three in total.

On question two, it was the sine curve that we're focusing on.

So, the first equation was between the x values were between 0 and 720, so there would be four.

If we go across at minus 0.

61 in this interval between 0 and 360, the one we can see, there are two solutions.

Then it was restricted between 270 and 720.

Well, we know that there are two between 360 and 720 because that's a full period.

So, there are two solutions between 360 and 720.

So now we need to focus on the extra bit of 270 to 360.

And we can see that here on the graph, and there is only one solution.

So, in total, that is three.

And now if we look at solving that between minus 360, so going into the negative values and 360, how many solutions would there be? Well, that's still two periods.

There's this period that we can see on the graph here and the one before it.

Question three was about the tangent function.

Well, the tangent function doesn't have a period of 360 degrees.

Instead, it repeats itself.

Remember that's what it means by a period.

It repeats itself every 180 degrees.

So, for question part a, it was solved between 0 and 360.

So, we can just make use of the graph we can see for 1.

6.

So, if we go across at y equals 1.

6, if we draw a horizontal line, then we can see that it would intersect our graph tan twice.

Then it says, what about from 0 to 540? 0 to 360 is two periods of tan, and there were two solutions.

So, if we're now going from 0 to 540, there's an additional period.

There's an additional 180 degrees, which means there is an additional solution.

So, that's why there are three solutions.

And then lastly, we've got minus 90 to 720.

Minus 90 to 720 is four and a half periods.

0 to 720 is four periods.

480s make 720.

So, it'd be four repeats of the tangent graph.

Minus 90 to 0 is half of a period.

That's only got a 90-degree difference.

So, it would be four periods.

We haven't got five periods, we've only got four periods.

So, to summarise today's lesson on interpreting the trigonometric graphs, reading solutions from trigonometric graphs is the same as reading solutions from any other graph.

Due to the period, there may be more than one solution, so you do need to check that.

And it is possible to predict the number of solutions that exist within a given range.

So, thinking about the period will allow you to think about how many solutions there will be to the equation.

Really well done today, and I look forward to working with you again in the future.