Lesson video

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

So by the end of this lesson, you should be able to use the tangent ratio to find missing angles or missing lengths on a right-angled triangle.

All of these keywords you'll have met before in your learning, but I would suggest that you pause the video just to refamiliarize yourself before we move on and make use of them during today's lesson.

The lesson's gonna take three parts.

The first learning cycle is to formulize the tangent formula.

The second part is to find an angle using the tangent ratio.

And lastly, we're going to make use of the tangent ratio in all variety of questions.

Let's make a start at formalising the tangent formulae.

We know that these two triangles are similar because two of the angles in one triangle are equal to two angles in the other, and we can find the length of the side labelled x by using a ratio table, because if they are similar to each other, then one is an enlargement of the other.

They're in proportion with each other.

So here is our ratio table based on the adjacent of both of those right-angled triangles.

One has an adjacent of one centimetre and the other has an adjacent of 12 centimetres.

And so what has happened to get from 1 to 12? We've multiplied by 12.

The opposite of 0.

67 needs to be multiplied by 12 as well to get the value of x.

Another way to consider this is the ratio in the triangle.

So to get from the adjacent of 1 to the opposite of 0.

67, you've multiplied by 0.

67, and that same ratio needs to withhold in the larger of the two triangles, the one with the adjacent of 12.

So 12 times 0.

67 is another way of getting to the x, sort of using the ratio within the triangle as opposed to between the two triangles.

Irrespective of how you do that, so whether you multiply the opposite by 12, the sort of scale factor, or whether you multiply the 12 by 0.

67, you'll get the same answer of 8.

04.

And we can go on to generalise this process for any right-angled triangle that has an angle of theta, where the side opposite theta has a length of tan theta.

So we're using opposite and adjacent, that's the tangent function.

So here, we've got our two adjacents.

One has an adjacent of one, the other has an adjacent of a.

So for any triangle, we're generalising this.

That is a multiplication of a.

Well, within the first triangle, the ratio between the opposite and the adjacent, there's this multiplication or the multiplier is tan theta, and that will be the same on the second triangle as well.

So a multiplied by tan theta will give you the opposite, and that would be an expression for the length of the opposite for any adjacent.

So any adjacent, so we've called that a, and any angle of theta, then that opposite is a times tan theta.

So the length of the adjacent multiplied by the tangent of the angle gives you the opposite edge.

And that is the general form that that is always true for a right-angled triangle, that there is this proportionality and this relationship between the adjacent and the opposite edge for sum angle theta.

So we can use the ratio table to generate a formula that shows this relationship between that side that's opposite the angle of theta and the adjacent, and we can represent it in three different ways.

So our a, which is our adjacent length, multiplied by tan theta is equal to the opposite, the edge that is opposite the angle of theta.

Or we could start on the opposite.

The opposite edge divided by tan theta is equal to the adjacent edge.

And hopefully you can see how that is just a rearrangement of the first formula.

And lastly, the opposite divided by a will give you tan of theta.

So again, it's a rearrangement.

We can rearrange and derive two versions of the formula given one of the others by just performing a division.

So if we start with this one, which tells us that the adjacent multiplied by tan theta, because tan theta is this multiplier between these two edges, is equal to the opposite.

Well, if we would divide both sides by a, which is our adjacent length, then on the left hand side, they would cancel out.

That would be equal to one.

So we'd be left with just tan theta, and on the right hand side, it would be opposite divided by the adjacent.

So that is one of the formulas that we just saw.

If we start on that exact same formula again but this time divide by something different, so if we divide by tan theta on both sides.

Tan theta is just a value, it's our multiplier, it's our ratio between the two edges.

Then on the left hand side, tan theta divided by tan theta would be one, so that cancels through, and leaves us with just a.

And on the right hand side, it would have to stay as opposite divided by tan theta.

So the adjacent edge, the length of that adjacent edge is equal to the length of the opposite divided by tan of the angle theta.

And that is our second and third of our formulae.

And we can substitute values in any right-angled triangle into each of the versions that we've just derived.

And they represent the relationship between the side opposite the angle and the adjacent, but just in different ways.

So we've got that a multiplied by tan theta equals opposite, and for this right-angled triangle, our a is b, our adjacent is b, our angle, theta, is 67 degrees, and our opposite edge is 16 centimetres.

So that represents the relationship between opposite and adjacent using the function of tan.

On the second formula, our opposite is 16, our angle theta is 67 degrees, and our adjacent is b.

And lastly, opposite is 16, adjacent is b, and our angle theta is 67 degrees.

Those three are all equivalent to each other.

They're just different rearrangements of each other.

So one correct equation showing the relationship between the adjacent and the side opposite the 38 degrees is 20 multiplied by tan 38 degrees is equal to opposite.

And you can see that in the ratio table, and hopefully you can match it to the right-angled triangle diagram.

So which of these are also correct to show the relationship between the adjacent and the opposite side? Is it a or is it b? So pause the video whilst you make a decision, and then when you went to check, press play.

So if we divide both of those by 20, then 20 over 20 would cancel through to 1, so you're left with tan 38 degrees, and that would mean that it's opposite divided by 20.

You can use the ratio table to see that as well, that tan 38 would be opposite divided by 20, so if you work in the other direction on the ratio table.

So which of these correctly shows the relationship between the adjacent and the side opposite the 82 degree angle for this triangle? So again, pause the video, work through each of those and see if it is showing the correct relationship between the opposite and the adjacent using the tangent function.

Press play when you're ready to check.

So C is the sort of definition of tangent that 10, the adjacent, multiplied by tan of 82 is equal to the opposite, which is 70.

So that one is correct.

But actually, there are two other versions, two other rearrangements that are also equivalent to that.

So A, because if we divide both sides by 10, you would end up with that.

And D, because if you divide both sides by tan 82.

A similar question, different triangle.

So which of these correctly shows the relationship between the adjacent and the side opposite the angle theta for this triangle? So once again, pause the video, make your decisions, and when you're ready to check your answers, press play.

So this one is our sort of definition of tangent.

So 64, the adjacent edge multiplied by tan theta equals 32.

Which of the others are the rearrangements and therefore equivalent? Well, B is the only one there that's another rearrangement of that, and that's by dividing both sides by tan theta.

So each version of the formula gives a different part as the subject, and so therefore it's helpful to use the version of the formula where the subject matches what you would actually try to calculate.

So on this right-angled triangle, we're trying to calculate p, and therefore the top one would be the most useful, because p is the opposite edge and opposite is the subject of that formula.

And then we can just substitute what we have.

So we know that theta is 55 and we know that the adjacent edge is 85, which is our A.

So 85 multiplied by tan 55 degrees is equal to p, the opposite edge, and we can calculate that on our calculators to get that p would be 121.

39 to two decimal places.

So a check on that.

Which of these equations helps find the value of q most efficiently? So pause the video and then when you're ready to check your answer, press play.

It would be the middle one.

All of the others are correct equations, but the q is not the subject, so you'd have to then rearrange it to make q the subject in order to solve.

So it's most useful to use it when q is already the subject.

Hence find the side labelled q.

So use the formula, use your calculator, what is the value of q to two decimal places? Pause it whilst you're calculating and then press play to check.

So q is 48.

56 metres.

Onto task now about formalising the tangent formula.

So question one, you need to substitute the labelled sides and angles from each of your triangles into the two versions of the tangent formula.

For each one, choose the most appropriate to find the length of its missing side.

Pause the video whilst you do those two parts, and then when you press play, we'll move to question two.

Here's question two.

You need to match the triangle, so match the diagram of the triangle to the equation showing the relationship between its adjacent and the opposite side.

And then use the equation to find that length x on each of them to two decimal places.

So match up and then calculate.

Pause the video whilst you do that, and then press play when you're ready for question three.

Here's question three.

You need to find the missing side length to two decimal places in each of the parts by using the tangent formula.

So pause the video whilst you work through that, and then when you're ready for question four, press play.

Here is question four, and you need to find the length of the distances a, b, c and d, and round them to the nearest integer using this diagram.

It may be useful for you to draw this as four separate right-angled triangles.

Once you start calculating some of the values, make sure you draw it and add them to your diagrams. Press pause whilst you work through that question, and then when you're ready for the answers to this task, we're gonna go through them when you press play.

So question one, you needed to substitute the values from the right-angled triangle into the formula, and then use the most appropriate one to calculate the missing size.

So on the first half, a, the second version of the formula was the most useful because a was the subject.

So you could type onto your calculator 119 divided by tan of 81 degrees.

On the second half, on b, the first one was more appropriate to use because b was the subject in that rearrangement of the formula.

So two times tan of 49 degrees.

And b was 2.

30 to two decimal places.

Question two, you needed to firstly match them, so a matched with e, b matched with f, and c matched with d, and then use them to calculate the value.

So just pause the video to check your answers, and then when you're ready to go to question three, press play.

Question three, there were six questions where you needed to use the tangent formula to calculate the missing edge length.

So again, I would pause the video and go through them one by one to check that you've got those correct, rounding to two decimal places, and then press play for the answers to question four.

Question four, a, b, c and d were missing lengths that you needed to calculate round to the nearest integer.

Unfortunately with this question, if the very first one potentially went wrong, then that would've affected or could have affected later answers.

So this second learning cycle is calculating the angle with the tangent function.

So on the screen, we've got the table of trigonometric values.

We've got the sine column, the cosine column, and the tangent column.

But we are just focusing on the tangent.

We know we're focusing on the tangent because this triangle here has the opposite and the adjacent, and there is this relationship that we just established between those two edges.

So tan of theta is equal to opposite divided by adjacent.

So using the right-angled triangle we have here, we can substitute our opposite, 0.

7, and our adjacent of 1.

So what is the size of the angle theta given the ratio between the opposite and adjacent as 0.

7 to 1? We need this relationship of 0.

7 to 1.

Well, it would be 35 degrees.

And so on the table of trigonometric values, that decimal value is the ratio between the opposite and the adjacent.

So here, we've got a different right-angled triangle.

We know it's tan because it's the opposite and the adjacent.

So if I substitute those into my fraction, we get 2.

184 divided by 6, which when we evaluate that gives you a value of 0.

364.

So on our table of trigonometric values, we are looking for that decimal, 0.

364.

So which angle would give a value of 0.

364 for tan of theta? And that is 20.

So we're going down the tangent column until we find the decimal and then reading off the angle.

So a check for you to do.

Which angle would give a value of 0.

466 for tan theta? Press pause whilst you identify that in the table, and then press play to check.

So it'd be 25 degrees.

So we found our decimal of 0.

466, and then that was 25 degrees.

So continuing here, substitute our opposite, substitute our adjacent.

Note that our opposite is now larger than our adjacent.

We've got a top heavy fraction.

The tangent function can give you results that are greater than one.

So that results is 3.

732.

So which angle would give you a value of 3.

72 for tan theta? Again, go to the table, identify in the tangent column, and then map it across to the angle.

So an angle of 75 degrees.

Aisha wants to know, "But how do we find angles" and hopefully you were considering this too, "that are not in the table?" So that table is going up in increments of five degrees, it's the multiples of five.

But we know that our protractors and any turn can be, are not always in multiples of five.

And so you can use the unit circle to find out some more angles.

So we've seen this diagram previously.

So here, if I'm trying to use this unit circle with my x equals 1, I've tangent drawn at the point x equals 1.

So the way that we're going to use our tangent function on this unit circle is we would firstly calculate that value, that ratio between the opposite and the adjacent, and that is gonna be our vertical height, that pink dashed line.

From there, we can draw our hypotenuse of our right-angled triangle down to zero.

We can trace across to our vertical axis, and that is the value that we would look up in our tangent function.

So 0.

577 would mean that this would be 30 degree angle.

The hypotenuse with the horizontal has made an angle of 30 degrees.

So here, if you've got this right-angled triangle, your first thing you're gonna need to do to be able to use the unit circle is get that value of tan theta.

So tan theta is the opposite divided by the adjacent, which is 0.

417.

So we're going to go up along our tangent to a height of 0.

417, and then draw on the hypotenuse from that point of 0.

417 down to zero, and that would be our angle theta.

And we can then use our circle to measure an estimate of what that angle is, which is approximately 22.

5.

We can see it's between 20 degrees and 30 degrees.

It's lower than the 25 degree mark.

So we can approximate that to 22.

5.

Going through that again here with this triangle, we've got an opposite of one and an adjacent of 1.

4, which is evaluated as 0.

714.

So we are going to go up the tangent to a height of 0.

714, and from that point, we're going to draw on our hypotheses.

So this is the angle theta that we're trying to calculate or estimate, and we can then read it, and that's 35.

5 approximately.

So here's a check for you.

If we have tan theta is equal to 0.

833, what is an approximate value for the angle theta? Pause the video whilst you're reading that off, and then when you want to check, press play.

So approximately 39.

8 degrees.

You can clearly see that it's not 40 degrees, so you shouldn't have put 40 degrees.

Approximately 39.

8.

So a value very close to but not exactly 40.

So Aisha has said, "Okay, but this method isn't very accurate." So we've managed to find angles that were not multiples of five, but they have been an estimation, so it isn't the most accurate way to get an angle.

And there must be another way that doesn't rely on the table of values.

How large would that table of values be if it was for every degree and every increment of a degree? Or a graph, and that is more accurate.

So the tangent function returns the ratio of opposite divided by adjacent for a given value of theta.

It's the multiplier between those two edge lengths on a right-angled triangle.

So if you can find the ratio from the angle, it is possible to use the inverse and find the angle from the given ratio.

And that's what we were doing on both the table of trigonometric values and also the unit circle.

But our calculator means we can do this quickly and more accurately.

So the tangent function is like an operation to find the ratio from the angle, and therefore the inverse of the tangent function finds the angle from the ratio.

So if we've got tan of theta is 0.

714, so that's our ratio, 0.

714, we want to use the inverse tangent function to find the angle of theta.

And so theta degrees would be equal to the inverse tangent function of 0.

714, and this has its own notation.

So we don't write inverse tangent function of our ratio every time.

We have two different notations, tan minus one of 0.

714, and that's the one you're gonna see on your calculator screens.

Or arctan is the vocabulary to mean inverse tangent function.

So it's programmed onto your scientific calculator.

It'll come up and it will look something like this on your screen.

You need to make sure that you have found the inverse tangent function on your calculator, and that's normally by pressing the shift or the second function level key before you press the tan button.

And then this will come up with the notation where you type the ratio in the brackets.

It's called arctangent for full or arctan, and that's its formal mathematical name.

And this, as you can see on the calculator, is 35.

52683659 degrees to eight decimal places.

So you can see that this is already more accurate than reading from the unit circle or using our tangent trig values.

And we can round that to one decimal place.

So a check, which of the following are equivalent to theta equals inverse tan of 0.

382? Pause the video.

It might be that you need to go back and rewatch a little bit before you come back to this check.

But when you're ready to press play and check, please do.

So the top three are equivalent.

The first one is the solution, the answer.

So we've evaluated the inverse of tan of 0.

382.

Part B is using arctan instead of the notation tan to the minus one.

Part C would be the line of working before, and part D is just incorrect.

So we're onto the task where you need to calculate angles.

So question one is about using the table of values to work out the missing angle.

So work out the multiplier, the decimal, go to the tangent column, and then map it across to find the angle.

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

Question two, I want you to do it by drawing appropriate lines onto the graph to estimate the angle in each triangle.

So these are going to be estimations, they're not gonna be really accurate, but you need to draw the appropriate lines onto the diagram, so that you can then estimate the angle.

Press pause whilst you do that, and then when you press play, you've got question three.

Question three, you're gonna now use that calculator to be as accurate as possible, but give your answer to one decimal place.

So pause the video, work through those four parts, and then when you're ready to go through the answers to questions one, two, and three, press play.

Question one, you needed to use the table.

So A was 40 degrees, B was 60 degrees, C was 45 degrees, and D was 20 degrees.

Opposite divided by the adjacent would've given you the decimal value to find on that tangent column.

Question two.

A was 38.

7 degrees approximately, and B was 45.

6 degrees approximately.

You may have a slightly different answer to what the ones I've written there, because we are estimating off of that diagram.

Question three, round your answers to one decimal place.

Part A, the angle was 38.

7 degrees, part B was 70.

2 degrees, C was 45.

6 degrees, and D was 26.

0 degrees.

So we're now going to use the tangent ratio in a variety of different questions.

Izzy and Aisha are discussing the meaning of tan of 36.

9 degrees equals three quarters.

Izzy says, "It applies to this triangle," and you can see her triangle there where she has an angle of 36.

9 degrees, her opposite is three and her adjacent is four.

It's a right-angled triangle.

Aisha said she actually thinks it applies to any right-angled triangle, where the opposite and the adjacent are in a ratio of three to four.

Who do you agree with? So actually, they are both correct.

What's the difference is is that Izzy is given an example of a triangle with that ratio.

It's not the only triangle.

So all of the triangles on the screen here are true for that statement that tan of 36.

9 degrees is equal to three quarters.

Within the triangle, there is this proportionality, this ratio of the adjacent multiplied by three quarters gives you the opposite.

And we can write them using the formula for tan as tan of the angle, so 36.

9 degrees is equal to the opposite divided by the adjacent.

And we can see that those opposites and adjacent fractions all simplify to three quarters, and that's why they work for all of them.

So a check, which of the following is the odd one out? So pause the video, think about what we just saw on that last screen, and then when you're ready to check, press play.

The middle one is the odd one out, and there's a few ways that you could come to that conclusion.

But one way is by looking at the ratio between the opposite and the adjacent, and the multiplier between them.

So on A and C, you multiplied the adjacent by 0.

7 and they gave you the result of the opposite.

Whereas on triangle B, you needed to multiply the opposite by 0.

7 to get the adjacent.

So a gradient of a line segment can be described as a ratio as well.

So change in the y-direction, so that's the vertical change, to the change in the x-direction, which is the horizontal change.

So here we have a line segment on a coordinate axis, and the line segment has a gradient of two, so that's our change vertically, change in the y-direction to four, and that's our change in the x-direction.

That's a ratio that simplifies to one to two.

Another way that we could describe the steepness or the gradient of the line segment is the angle that it makes with the horizontal.

So you can see that marked there on the diagram.

And it can be calculated using the tangent function, because we know the change in the horizontal and the change in the vertical.

We know the opposite edge and the adjacent edge.

So tan of this angle theta is two over four, which simplifies to 1/2.

So we saw the gradient as a ratio two to four, which simplified to one to two.

And here we've got the ratio written as a fraction, so 2/4, 1/2.

And we can use our arctan to calculate that that angle is 26.

6 degrees to one decimal place.

And what this means is that for any line segment that has a gradient of one to two, then the angle it makes with the horizontal is 26.

6 degrees.

Wheelchair access ramps in public buildings have regulations regarding the gradient.

The gradient must not exceed 1 to 12.

So what is the maximum angle of incline that a wheelchair access ramp can be? So here is a sketch of a right-angled triangle.

And so our angle theta is that angle of incline, the x and the 12x is the ratio of the gradient, the change in the y-direction to the change in the x-direction.

So tan of theta is equal to the opposite, divided by the adjacent, and so x divided by 12x simplifies to 1/12.

And using arctan on 1/12 results in the angle 4.

76 degrees to two decimal places.

So here is a check for you.

The incline of a wheelchair ramp cannot exceed 4.

76 degrees.

If a ramp was installed with a gradient of 20 to 240, would the incline exceed 4.

76 degrees? Pause the video whilst you calculate that, and then when you're ready, press play to check.

So no, it would effectively be at its maximum.

That ratio 20 to 240 simplifies to 1 to 12.

So we don't need to use trigonometry, but you could.

But that is in the same ratio, so if it's at the same ratio, then it's at that maximum angle of incline.

If a ramp was installed with a gradient of 10 to 130, would the incline exceed 4.

76 degrees? So once again, pause the video, and then when you're ready to check, press play.

So no, this one would be less than the maximum.

The run of the slope, so the horizontal change is more than it needs to be, and that would therefore decrease the angle.

So we're up to the task where you're going to use the tangent ratio.

So question one, using tan theta equals three over five, I'd like you to complete the edge lengths of these six triangles.

Pause the video whilst you do that, and then when you press play, we'll move to question two.

Here's question two.

You need to calculate the angle that each line segment makes with the horizontal.

So press pause whilst you work through those two parts, and then when you're ready for question three, press play.

So we've got some more wheelchair access ramps being installed at various public places, and I'd like you to work out the required distances for each of the ramps.

So pause the video and work through those four parts.

And then when you press play, we're gonna go through all the answers to this task.

So question one, you needed to fill the edge length so that tan theta was equal to three over five.

So we needed to keep this ratio between the opposite and the adjacent constant as 3/5.

So on part A, the opposite was three, therefore the adjacent needed to be five.

On part B, the opposite was 30, so the adjacent needed to be 50, because if you do the opposite divided by the adjacent, it would simplify to 3/5.

So you may wish to pause the video so you can work through those answers and check against your own.

Question two, you needed to calculate the angle that each line segment made with the horizontal.

So by drawing on a right-angled triangle to identify the lengths of your opposite and adjacent, you then could use arctan to calculate that angle.

So on part A, it was 21.

8 degrees to one decimal place, and on part B, it was 63.

4 degrees to one decimal place.

And finally on question three, you need to work out the distances on each of those wheelchair ramps.

Again, you may wish to pause the video, so that you can work through each of those and check against your working out.

So to summarise today's lesson, which was using the tangent ratio.

The tangent ratio involves the opposite, the adjacent and the angle.

The relationship between these can be expressed in three different ways, and so we can rearrange the formula to give us different subjects.

This tangent formula can be used to find the length of the opposite, that's the first one, the adjacent, which is the middle one, and the angle of theta, which is the last one.

So in order to find the size of the angle, we need to use the inverse function, and that's arctangent.

On your calculator, the arctangent function is usually written as tan to the power of minus one, and you find that by pressing your shift key.

Really well done today.

I look forward to working with you again in the future.