Lesson video

Hello.

I'm Mrs. Lashley and I'm going to work with you as we go through the lesson today.

I really hope you're looking forward to the lesson, you're willing to give it your best shot, and I will be there to support you as we get through it.

So the learning outcome today is to be able to use the cosine ratio to find the missing side or angle in a right-angled triangle.

So there's some keywords that I'll be using during the lesson that you will be familiar with in your previous studies, but you may wish to pause the video just to make sure you're feeling confident before we make a start.

So our lesson on checking and securing understanding of cosine problems is gonna be split into two learning cycles.

The first learning cycle, we're gonna look at calculating an unknown length, and then later on we'll get to the second learning cycle, we'll be calculating the angle.

So let's make a start looking at reminding ourselves how we've use the cosine ratio to find lengths.

So we can use this ratio table to generate the formula that shows the relationship between the side adjacent to the angle theta and the hypotenuse, and represent the formula in three ways.

So we've got the diagram of a right-angled triangle.

We know the hypotenuse is the longest edge opposite that right angle.

And then the adjacent is the edge between the angle theta and the right angle.

And we can see from the ratio table that the adjacent has an expression of hypotenuse multiplied by cosine of theta.

So it's the product of the cosine of theta and the hypotenuse length.

So if we look at the hypotenuse and come up with a formula that involves the hypotenuse, the opposite, the hypotenuse, the adjacent, and the angle theta.

So using the ratio table, we can see that the hypotenuse multiplied by cosine theta gives us the length of the adjacent.

And remember that adjacent is the edge between the angle theta and the right angle.

So that's one form of the cosine formula.

If we start with the adjacent, so the edge between the angle theta and the right angle, well what does the ratio table show us? Well, the ratio table shows us that if we divide the adjacent by cosine theta, then that will give us the length of the hypotenuse.

And so that is a different form of the cosine formula.

And finally, if we start with the adjacent again, so the length between the angle theta and the right angle, and this time divided by the hypotenuse, we can see that this is equal to the cosine of theta.

So here is the cosine formula in three different ways.

The subject is different in each one of them.

So we can substitute values from a right-angled triangle into each version to represent that relationship between the side adjacent to the theta and the hypotenuse.

So here we've got our right-angled triangle.

Our angle theta is 72 degrees, our hypotenuse has a length of b, and our adjacent has a length of 21 centimetres.

So if we substitute it into the top form, then we'll have the hypotenuse, which is b, multiplied by cosine of 72 because the angle theta is 72, is equal to our adjacent length of 21.

Or if we substitute those values into the relevant places into the second form, we'll have 21 divided by cosine 72 degrees is equal to b where b is that hypotenuse.

And lastly, 21 divided by b is equal to cosine 72.

So these three equations are equivalent to each other.

They're just rearranged forms. So in each version of the formula, the subject is different and it's helpful to use the version of the formula whose subject matches the part of the triangle that you are trying to find the value of.

So in this example here, we have the hypotenuse of 103 centimetres, We have theta at 57 degrees, and the adjacent is the unknown length.

That is what we're trying to work out.

So which of the three forms has got the adjacent as the subject? Well, it's the top one.

So this would be the most relevant form of the cosine formula to make use of.

We can substitute in our values.

103 is our hypotenuse, 57 is our theta.

And then we can see that our p, our adjacent length, is just the product of 103 and cosine 57.

We can use our calculator to work out the answer.

So the answer, p, which is the adjacent, so less than the hypotenuse, that's what we expect, is 56.

098 to three decimal places.

So when given an angle and the adjacent side or hypotenuse, it's possible to determine the remaining side.

So because if we think about the formula, it involves three variables.

It involves theta, the hypotenuse, and the adjacent when we are using cosine.

So if we have the hypotenuse of 250 centimetres and we have the angle theta of 75 degrees, then it's possible to work out a, the adjacent.

So if we substitute those parts in and calculate, then we get our length of our adjacent for this example.

Here's another example where we've got the adjacent side this time, the angle, but it's the hypotenuse that we're trying to calculate.

So this form of the formula is the one we're going to use where h, the hypotenuse, is the subject.

So we're gonna substitute our values into the relevant parts of the three variables.

We have two of them.

So we can calculate the third.

So the numerator is the adjacent, which is 15.

Theta is 47.

And so on our calculator we can calculate what 15 divided by cosine of 47 degrees is.

And that will give us the length of our hypotenuse which we have labelled as b.

So what's the value of b? Well, as I say, use your calculator and calculate that, b is 21.

99.

That's rounded to two decimal places.

So here is a check for you.

Which of these calculations is correct? Pause the video, and when you're ready to check, press play.

So a is correct.

A, 250 centimetres is the hypotenuse, the longest edge opposite the right angle.

Our angle theta is 75 degrees and we know that the adjacent is the product of the hypotenuse and cosine of the angle.

And we can see that the adjacent edge is marked as 250 multiplied by cosine of 75 degrees.

Whereas if we look at b, the edge that is marked is actually the opposite to the 75 degree angle.

So cosine would not be the correct trigonometric ratio to be using there.

And if we look at c, it is the adjacent that has got marked up, but that calculation is not correct to find the adjacent length.

So onto the first task of the lesson.

By using the cosine formula, find the length of the missing side labelled with a letter for each of these triangles and round your answers to two decimal places.

So you've got six triangles and you need to work out the missing side length that's labelled by the letter.

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

So the answers are on the screen.

We've got a as 0.

80 to two decimal places.

That's less than the hypotenuse and that's what we would expect.

On b, b is 17.

57.

Again, less than the hypotenuse of that triangle.

And we would expect that the hypotenuse is the longest length.

For c, it's 15.

00 to two decimal places.

D is the hypotenuse that you're trying to calculate.

So your calculation would've been different from a, b, and c.

You would've used the formula in a different rearrangement.

So d is equal to 32.

26 to two decimal places.

On triangle e, e is equal to 30.

08.

You may have used the sine formula.

The question did ask you to use the cosine formula, but you had sufficient information without having to work too much else out that you could have used the sine formula by using 22 as your opposite and 47 as your angle theta.

Whereas I'm hoping you do 22 as your adjacent and the angle 43 as your theta.

And for f, f is 50.

69 to two decimal places.

The adjacent was 47, the angle theta was 22.

So for a, b, and c, you should have been doing hypotenuse multiplied by cosine theta to work out your adjacent.

Whereas for d, e, and f where you were trying to calculate the hypotenuse, you should have been doing your adjacent length divided by cosine of your theta.

So we're now up to the second learning cycle where we're gonna still be working with the cosine ratio, but this time we're gonna be calculating an unknown angle.

So on a unit circle, we can find the length of a side adjacent to the angle 30 degrees by applying the cosine function to that angle.

Let's just bring that back slightly.

So a unit circle is our circle on a coordinate grid with a radius of one centred at the origin.

And here we can see the first quadrant, a sector of the unit circle.

So if we've got the angle of rotation of the radius from the horizontal axis to that point there of 30 degrees, then we can see that we've got this right-angled triangle.

And the right-angled triangle, if we're looking for the cosine of 30 degrees, then we are looking at the x coordinate.

So if we take our 30 degrees and we apply cosine to it, then we read off the x coordinate of the point to find the value.

And that's 0.

866 to three decimal places.

And you can see that that gives us that horizontal edge and that is our adjacent.

So the length of the adjacent side on the unit circle is what that's equivalent to.

The cosine function has an inverse function called arccosine or arccos for short.

And the arccos or the arccosine is going to be a really important factor for calculating the unknown angle.

When we use our calculator or also in any other text, you may see it written as cos and it looks like a power of -1 but it's actually the inverse function notation.

So that function notation is telling us that this is the inverse cos function or arccosine.

So if we apply the inverse function, then we end up saying that cos of 30 degrees is equal to 0.

866.

But if we apply the inverse function of cos onto this equation, then 30 degrees is equal to the inverse of cos on 0.

866.

So the length of the adjacent side of the unit circle, if we apply the inverse cos function to that, we get the angle that created it.

And our calculator has this function programmed onto it.

So if we've got this triangle here, it's a right-angled triangle, so we can use cosine.

We've got the hypotenuse of 89 centimetres, we've got that adjacent side to the angle theta which is 38 centimetres.

Well, when we put that into the cosine formula, we get cosine of theta is equal to the adjacent divided by the hypotenuse.

So 38 divided by 89.

But we want to find out what theta is, not cosine of theta.

So we're gonna use the inverse cosine function arccos.

So we need to type the left hand side of that equation onto our calculator and this is how you'll do it.

So you're going to press the shift key or that might say second function.

And then you're going to press the cosine button.

The cosine button on the second function is actually the inverse cosine function.

And you can see the notation on the screen of your calculator.

So now you're gonna type the ratio of adjacent and hypotenuse, and in this particular example, it's 38 divided by 89.

So press your fraction key, type your numerator, move to the denominator using the arrow keypad, and type your denominator, close your bracket, and then press execute.

And so this is returning to you the angle of theta such that the adjacent side length is 38 when the hypotenuse is 89.

And so to three significant figures, this angle is 64.

7 degrees.

So here is a check for you.

Which of the following equations are correct when finding the size of the angle n in the equation? So pause the video, have a look, and when you're ready to check, press play.

So it's part b and part c.

We need to use the inverse cosine function.

So the inverse cosine function may be written using inverse function notation like you can see in b, or it may be written as arccos.

Here's another check.

Which of the following equations are correct when finding the size of the angle u in this triangle? So pause the video, and when you're ready to check, press play.

So a and d.

So cosine of u is equal to the adjacent divided by the hypotenuse.

And then if we want to make u the subject, if we want to find the value of that angle, then we're gonna use our inverse cosine function.

And the fraction of the adjacent divided by the hypotenuse would stay with the numerator is 18 and the denominator is 60.

So we're now up to the last task of the lesson for you.

And so there's one question here and you've got six parts to it, a through to f.

So find the size of each angle marked with a letter and round your answer to one decimal place.

On part a and part b, there's a little bit of support, a little bit of scaffolding there to get you going.

And then for c, d, e, and f, hopefully you can use that as a structure and a support to work out the others.

So do make sure your calculator is in the degree mode.

You'll find a little D on the top of the screen to indicate that your angles are being measured in the unit of degrees rather than radians for example.

If your calculator is on the wrong mode, then your answer will come out incorrect.

So pause the video and then when you're ready to check all the answers to a through to f of this Task B, press play.

So all the answers are on the screen, but I'm gonna talk through them one by one.

So on part a, the working had already been done much of it and you needed to just type that onto your calculator correctly.

So the reason it's 20 divided by 40 is because the adjacent is 20 and the hypotenuse is 40.

And then you're gonna use arccos of 20 over 40, and that gives you the angle of 60.

0 degrees.

One decimal place was the degree of accuracy that you were supposed to be giving your answers, and therefore you did need a.

0.

On part b, you needed to put the numerator and the denominator into that fraction.

The numerator is the adjacent, the denominator is the hypotenuse, so 15 over 40.

If you chose to simplify that fraction because it does simplify, then you'll get the same answer.

But because you're going to use your calculator anyway, there's not much of a need to simplify the fraction because the calculator will understand that 15 over 40 is the same as 3/8.

So the answer, the angle is 68.

0 degrees.

Then if we go to c, we've got a hypotenuse of one centimetre and an adjacent edge of 0.

5 centimetres and the answer is the same as part a.

Well that's because if you have a look, those two triangles are similar.

So we would expect the angles to be the same.

If you look at the ratio between or the proportion of the adjacent to the hypotenuse, it is half the length and that's the same on c.

And that's the reason that the angle is the same, because the proportion of the adjacent to the hypotenuse is the same.

If we look at d, we've got an adjacent edge of 0.

5 and a hypotenuse of two.

The angle is not twice the size as c.

The adjacent has stayed the same, but the hypotenuse has doubled.

That doesn't mean that the angle will double.

So the cosine function is not a linear function, so it doesn't work in that same way.

So the angle is 75.

5 degrees to one decimal place.

If we have a look at e, the angle is 48.

2 degrees to one decimal place.

The adjacent was two and the hypotenuse was three.

And lastly we're up to f.

And f has an angle of 36.

9 degrees to one decimal place.

If you came across when you were working any of these questions out your calculator saying math error, the reason that will be is because you had your fraction the wrong way up.

You had it inverted.

If you had tried to input arccos of 40 over 15, for example, it would return math error because the cosine function will never have a value greater than one and 40 over 15 has a value over one.

So it's important that you make sure that you are considering what the ratio is, which is the adjacent divided by the hypotenuse, and therefore your decimal value of that fraction would be less than one.

So to summarise today's lesson on checking and securing understanding of cosine problems, the cosine ratio involves the hypotenuse, the adjacent, and the angle.

If you know the length of the hypotenuse and the size of the angle, then you can use the cosine ratio.

If you know the length of the adjacent and the size of the angle, you can use the cosine ratio.

And if you know the lengths of the hypotenuse and the adjacent, you can use the cosine ratio.

So the cosine ratio, there are three forms of the ratio and the formula.

Each one has a different subject and it's for different things.

So to find the adjacent length, you'd use the one where the adjacent is the subject.

To find the hypotenuse, you'd use the formula where the hypotenuse is the subject.

And to find the angle, you'd use the formula where cosine theta is the subject.

And then we'd make use of arccosine, the inverse cosine function.

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