Lesson video

Hello, I'm Mrs. Lashley and I'm gonna be working with you as we go through the lesson today.

In today's lesson we're gonna have worked out how to use the cosine ratio to work out missing angles as well as missing side lengths in a right-angled triangle.

On the screen, there are some keywords that we'll be using throughout the lesson.

It might be that you wish to pause the video so you can re-familiarize yourself with those definitions before we start using them during today's lesson.

The first learning cycle is to formulize the cosine formula.

The second learning cycle is to calculate angles using that formula.

So let's make a start at "Formulizing the cosine formula." So we know that these two triangles are similar.

The reason we know that is 'cause two of the angles in one of the triangles is equal to two angles in the other.

And we can find the length of the side labelled x by using a ratio table.

So these two hypotenuse, there's a 1-centimeter hypotenuse on one of the triangles, and then the similar triangle has a 15-centimeter hypotenuse.

And so we can see that that has been multiplied by 15.

The scale factor of the enlargement of these similar triangles is 15.

The 0.

68 is the adjacent of the smaller of the two triangles.

And internally we can see that the hypotenuse multiplied by 0.

68 gives you 0.

68, so the proportion between those two edge lengths is 0.

68.

And that would be the same in our similar triangle.

To find the edge marked x, we can do 0.

68 multiplied by 15, and that is 10.

2.

So we now know that on a similar triangle to the original one, the adjacent would be 10.

2 centimetres.

And we can go further to generalise this for any right-angled triangle that has an angle of theta where the side adjacent to the theta and the right angle has a length of cos theta.

Back to our ratio table, we've got a hypotenuse of 1 and a hypotenuse that we've labelled as h because this is our generalised triangle, so that h could be any number.

The scale factor of the enlargement is h.

1 multiplied by h gives you the h.

And therefore, cosine theta multiplied by h would also give you this adjacent length on the larger of the two triangles.

The ratio in the triangle is that the 1, the hypotenuse of 1 multiplied by cosine theta gives you cosine theta.

And that would be the same on the second triangle, that the hypotenuse h multiplied by cosine theta is the length of the adjacent, which we could write as h times cosine theta, an expression for the edge length.

What does this tell us? Well, it tells us that the length of the hypotenuse h multiplied by the cosine of the angle theta equals the length of the side adjacent to that angle and the right angle.

And this is true for any right-angled triangle.

So using the ratio table, we can generate a formula that shows this relationship between the adjacent side and the hypotenuse, and actually we can represent that in three different ways.

So let's look at that now.

If we take a hypotenuse of any right-angled triangle and multiply it by cosine theta, then that would calculate or be equal to the adjacent edge.

So that's one form, that's one formula that shows the relationship between the adjacent side and the hypotenuse.

If we start with the adjacent, then working with an inverse, we can divide by cosine theta and that will be equal to our hypotenuse.

So this is a second form of the formula.

And lastly, if we start at the adjacent again, we can divide by the hypotenuse and this will be equal to cosine of the angle theta.

So here are three versions of the formula.

We can also derive the two other versions by doing a division.

So here is that first one that we formed.

The hypotenuse multiplied by cosine theta is equal to the adjacent.

If we divide both sides of the equation by the length hypotenuse, then on the left-hand side we'll end up cancelling out the hypotenuse, that will be equal to 1, and so all that will be left on that side is cosine theta.

On the right-hand side, there is nothing to cancel down, so we can say that cosine theta is equal to the adjacent divided by the length of the hypotenuse.

If we start with that same formula again but this time divide by cosine theta on both sides, on the left-hand side, the cosine theta divided by cosine theta will cancel down to 1, so we'll now have that the hypotenuse is equal to the adjacent divided by cosine theta.

And so we've derived and come up with the same three versions of that formula.

When we're working with a non-generalized triangle, so this one here we've got a hypotenuse of b, an adjacent of 21, and an angle theta of 72 degrees, we can substitute those into these three formulas.

b times cosine 72 degrees equals 21.

Or 21 divided by cosine 72 equals b.

Or 21 divided by b equals cosine 72.

We've just substituted the relevant part into our generalised formula.

So here is a check.

One correct equation for the right-angled triangle shown here is 20 multiplied by cosine 38 degrees is equal to the length of the adjacent.

So which of these are also correct to show the relationship between the hypotenuse and the adjacent side? Pause the video whilst you decide between A and B, and then when you're ready to check, press play.

So dividing both sides by 20 will lead us to the formula or the equation B.

Cosine of 38 degrees is equal to the adjacent divided by 20.

So another check for you.

Which of these correctly shows the relationship of the hypotenuse and adjacent for this triangle? So pause the video whilst you decide which of those five are showing correct relationships.

Press play when you're ready to check.

So if we start at C, C is correct.

This one shows us that the hypotenuse multiplied by cosine of the theta is equal to the adjacent.

B and E are also correct equations.

So which of these correctly shows the relationship between the hypotenuse and the side adjacent to the angle theta for this triangle? Again, pause the video, and once you decide, press play so that you can check.

D is in that original form of hypotenuse multiplied by cosine theta equals adjacent.

And B and C are rearrangements of that.

So each version of the formula, why do we have three different versions? Well, that's because a different part of the triangle is the subject.

And so it's helpful to use the version where the subject matches the part of the triangle that you are trying to find the value.

So if you look at this right-angled triangle here, we are trying to calculate what the length of p is.

Labelling the triangle, we can see that p is the adjacent, so we would want to use the formula where p, or the adjacent, is the subject.

Which is the top one.

So we are going to use this form because the adjacent is the subject.

We're gonna substitute in our values.

103 is our hypotenuse length, the angle theta is 57 degrees, and we can then calculate that on a calculator.

And p is 56.

098 centimetres to three decimal places.

So another check.

Which of these correctly shows a relationship between the hypotenuse and the side adjacent to the angle of 16 degrees for this triangle? Again, pause the video, and when you're ready to check, press play.

So A, B, and D are correct for this triangle.

Which of these is the correct equation to find the value of q most efficiently? So they are the three that you just chose, A, B, and D.

Which one is the most efficient to use if you are trying to calculate the value of q? Pause the video while you're deciding and then press play to check.

q is the subject on B, so B is the one that you would use to be the most efficient.

So find the value of the side labelled q to two decimal places.

Pause the video whilst you use your calculators to calculate that and then press play to check your answer.

So to two decimal places, it is 88.

43.

This seems like a very sensible answer since this is the hypotenuse and it should be greater in length than the 85.

So it's always worth to do a sense check just to check that the value has come out either greater or less than, depending on what you expect.

So we're up to the first task of the lesson, and on question 1 you need to substitute the labelled sides and angles in each triangle into the cosine formula and find the length of its missing side.

Press pause whilst you're doing that, and when you press play, we'll go on to question 2.

Here's question 2.

You need to firstly match the triangle to the relevant equation, and then use that equation to get the answer to two decimal places.

So press pause whilst you're matching and calculating, and when you press play we'll move on to question 3.

Question 3, there are six right-angled triangles for you to use the cosine formula to find the lengths of the missing labelled sides.

Again, give your answer to two decimal places.

Press pause whilst you're calculating those missing edge lengths, and then when you come back, there'll be question 4.

Question 4, you need to find the lengths of the distances marked A, B, C, and D on this diagram, rounding your answers to the nearest integer.

It might be that you want to draw these as separate triangles to help you and anything you do work out in a previous part move on to the next diagram.

Press pause whilst you're working through question 4, be very careful and round at the very final stage of your calculation.

When you press play, we'll go through the answers to Task A.

So question 1, you needed to substitute from the right-angled triangle diagrams into the formula and then solve it to find the value of a and b.

So the value of a was 25.

19 and the value of b was 5.

56.

Question 2, you needed to match and then calculate.

So the triangle A matched with the equation D, triangle B matched with the equation F, which left triangle C with E.

Then using a calculator, you can find the value of x giving your answer to two decimal places.

Question 3, there were six right-angled triangles and you needed to find the length of the missing side which was labelled with a letter for each one, giving your answer to two decimal places.

It may be that you wish to pause the video so that you can go through them one by one and check your answers.

Question 4, a was 34 centimetres, b was 12 centimetres, c was 16 centimetres, and d was 5 centimetres.

With this style of question, unfortunately, if one of your first answers had an error, that may have moved down the rest of your working.

So it might be that you find that you've got a correct but you've got b incorrect and then that means that c is incorrect and d is incorrect.

So I would pause the video if you have made an error and re-go through your working out.

The second learning cycle is "Calculating angles using the cosine formula." So we've been working with lengths and now we're moving on to angles.

Jacob and Izzy have both been saying that they can calculate the lengths of sides because they have an angle and they also have one of the edge lengths.

So Jacob, "I can calculate the length of the side adjacent to the 40-degree angle because I also know the length of the hypotenuse." And Izzy said she can calculate the length of the hypotenuse because she knows the length of the side adjacent to the 59 degrees.

So here is Jacob's example.

If he wants the adjacent, he knows that 6 times cosine 40 will give him the adjacent, and that is 4.

60.

For Izzy, she's trying to calculate the hypotenuse, she says she can do that with the length of the adjacent.

She would use this form of the formula because she is trying to calculate the hypotenuse.

So the hypotenuse is 3 divided by cosine 59 and that is equal to 5.

82.

Jacob is wondering, "How do we use this form of the cosine formula to find the angle if we know the length of the hypotenuse and the adjacent to that angle?" We can substitute in those two values, we can substitute the adjacent and the hypotenuse, which is approximately 0.

565 when you evaluate that fraction to a decimal.

So how can you find the size of the angle theta? Well, 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.

So our angle is 30 degrees.

And if we apply the cosine function to it, it gives us a value of 0.

866.

And that is the value along the X axis, so that's 0.

866 units.

And that's the length of the adjacent side on a unit circle.

The cosine function has an inverse function called arccosine or arccos for short, and you may see it as cosine to the power of -1 on your calculator.

So how we apply this is if we do arccos to cosine of 30 degrees equals 0.

866.

That's an equation, so we're gonna do it to both sides of the equation.

Then the length of the adjacent side on the unit circle was 0.

866, and that gives us a 30-degree angle.

So finding the cosine of an angle outputs a ratio of the side adjacent to that angle and the hypotenuse, and that's usually written as a decimal with respect to a hypotenuse of 1.

We can interpret a decimal as a set of equivalent fractions.

So cosine 30 degrees is 0.

866/1, which is telling you that the adjacent is 0.

866 in length on a triangle with a hypotenuse of 1.

But this is equivalent to 1.

732 divided by 2, which is a triangle where the adjacent of 1.

732 has a hypotenuse of 2.

Or even 86.

6/100.

This right-angled triangle would have an adjacent of 86.

6 and a hypotenuse of 100.

So the inverse function, the arccos function, takes a ratio of the side adjacent to the angle and the hypotenuse and gives you the size of the angle.

So arccos of 13/23.

Well, this is a triangle where the adjacent has a length of 13 and the hypotenuse has a length of 23, which we can evaluate as a decimal instead, and would compute, would give out an output of 55.

6 degrees.

So we would now know that a right-angled triangle with an adjacent and a hypotenuse of 13 and 23 respectively have an angle of 55.

6 degrees between them.

And we can use our calculator to find this angle.

So here we've got another example of a right-angled triangle.

We've got an adjacent between theta and a 90-degree of 38 centimetres and a hypotenuse of 89 centimetres.

So cosine theta is equal to 38/89.

And using arccos we would work out the angle of theta.

So let's use our calculator to do this.

To write this onto the calculator, first of all you need to press the Shift key, the arrow.

Press the Cosine button and this will bring arccos onto your screen.

We're now up to the point of writing the ratio between the adjacent and the hypotenuse.

So press the Fraction key, type 38, move down using the Arrow key, type 89, and then press to the right so that you come out of the denominator, otherwise your cursor is still on the denominator.

And then close the bracket.

Press Execute.

This has now returned the angle of 64.

72 degrees, or 64.

7 degrees to one decimal place.

So here is a check.

Which of the following equations are correct when finding the size of angle n in the equation? So cosine of n degrees is equal to 0.

86.

Which of the four are correct? Press pause whilst you work through that, and when you're ready to check, press play.

B and C are correct.

So B is using the notation that you see on your calculators, so cosine with like a power of -1.

And part C is using arccos which stands for the inverse cos.

Another check, which of these are equivalent to this equation? Once again, pause the video and then when you're ready to check your answers, press play.

A, C, and D.

So A is equivalent because 8/18 is equal to 4/9.

4/9 is the simplest form of 8/18.

C is using arccos on 4/9 and D is also using arccos on 4/9, except 4/9 has been written as an equivalent fraction of 40/90.

Another check, which of the following equations are correct when finding the size of angle u in this triangle? A, B, C, or D? Pause the video whilst you make your decisions and then when you're ready to check, press play.

A and D.

So A is written as cos of u degrees is equal to the ratio of the adjacent and the hypotenuse, whereas D is solving to find u, so we're using the arccos function.

B is not using the arccos function, so that's not gonna calculate the angle.

And C, the fraction is the wrong way up, it's inverted, so we need to get adjacent divided by hypotenuse.

If you try to do C on your calculator, it will come up with a math error, and that is because the value of cosine will never be greater than 1.

69/16 is clearly a top-heavy fraction, an improper fraction, and therefore greater than 1.

So we're onto the task of finding angles using the cosine formula.

So question 1, you need to match the triangle to the equation and then to the size.

So there's three parts that you need to match up.

Press pause whilst you're doing that and then when you're ready for question 2, press play.

Question 2, there are six angles that you need to calculate using the cosine formula.

Part A and part B has got a little bit more structure for you, so you need to fill in the missing gaps, and then you can continue through C, D, E, and F.

Press pause whilst you're working through those six questions.

And then when you press play, we'll go through the answers to Task B.

So Task B, question 1.

You needed to match the triangle, the equation and the value.

So A, the adjacent was 3 centimetres, the hypotenuse was 4 centimetres, cosine theta was equal to 3/4.

When you do arccos of 3/4, you get 41.

41 degrees to two decimal places.

B, adjacent was 3, hypotenuse was 5, and theta was the angle you were trying to evaluate.

And that came out as 53.

13 degrees using equation F, so arccos of 3 divided by 5.

And lastly C, the adjacent was 4, the hypotenuse was 5, so cosine of theta is equal to 4/5.

And when you use arccos of 4/5, you get 36.

87 degrees.

Question 2, you had those six questions where you needed to find the angle in each of those triangles.

A and B had a little bit of structure for you.

So part A, 32.

0 degrees was the answer to the angle a.

Part B, 28/71 would've been the fraction, so adjacent over hypotenuse.

And then, solving that on the calculator, 66.

8 degrees.

C, 29.

5 degrees.

D, 64.

2 degrees.

E, 14.

8 degrees.

And F, 29.

5 degrees.

So to summarise today's lesson, the relationship between the hypotenuse of a right-angled triangle and the side adjacent to an angle theta can be written as a formula in three different ways, and they're there on the screen.

This cosine formula can be used to find the length of the hypotenuse, which is the middle version, the length of the side adjacent to the angle theta, which is the right-hand formula, or the size of angle theta itself, which is using the left-hand formula.

In order to find that angle, the function inverse to the cosine function must be used, and this is the arccos function.

On a calculator, we're gonna see arccos or arccosine usually written as cos to the power of -1.

A triangle can be constructed from its cosine formula and either the length of its hypotenuse or the side adjacent to one of its angles in order for it to be a unique triangle.

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