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, our learning outcome today is to use the tangent ratio to find the missing side or angle in a right-angle triangle.

So, on the screen, there are some keywords that we'll be making use of during this lesson.

They're not new keywords, you have met them before in your learning, but let's just take a look at them now.

So, trigonometric functions are commonly defined as ratios of two sides of a right-angle triangle containing the angle.

The tangent of an angle is the y-coordinate of point Q of the triangle, which extends from the unit circle, and you can see that on the diagram.

The adjacent side of a right-angle triangle is the side which is next to both the right angle and the marked angle.

The opposite side of a right-angle triangle is the side which is opposite the marked angle, and again, those diagrams have got them labelled up.

So it might be that you want to pause the video here, read that again to yourself, check the diagrams, and that you're following with those definitions, but as I say, we'll be using those in the lesson.

This lesson is checking and securing understanding of tangent ratio problems, and we're gonna split the lesson into two learning cycles.

The first learning cycle is revisiting the tangent formula to find sides.

So, we're gonna make use of the tangent ratio to find edge lengths of a right-angle triangle, in particular the opposite and the adjacent.

And then when we get to the second learning cycle, we're gonna revisit the tangent formula to find angles instead, and that angle is going to be the one that is relative to the opposite and the adjacent.

So, let's make a start at finding the lengths of right-angle triangles by making use of the tangent formula.

So here on the screen we've got the three versions of the tangent formula.

We've got a, which is standing for the adjacent, multiplied by tan theta, where theta is the given angle and that will give you the length of the opposite.

So, the product of the adjacent and tan theta gives you the opposite.

A rearrangement of that, and hopefully you can see how that's being rearranged.

If you divide the opposite by tan theta, then that will compute the length of your adjacent, and finally, tan theta is equal to the length of the opposite divided by the adjacent.

Here we can see that the ratio of the two edge lengths, tan theta, is a ratio between the opposite and the adjacent, and so we can substitute values in a right-angle triangle into each version of the tangent formula to represent that relationship between the side opposite the angle and the adjacent in different ways.

So, if we use this particular right-angle triangle, we know that the angle theta is 60 degrees, the opposite side to that 60 degrees is nine centimetres in length, and the adjacent to that 60 degrees is b.

We haven't got the value of that length; we've labelled it as b.

So, when we substitute that into the three forms, we'd have b multiplied by tan 60 degrees is equal to nine.

Nine divided by tan 60 degrees is equal to b, and nine divided by b equals tan of 60 degrees.

So, if we were looking to calculate the value of b, which of the three forms would be the one to use? They're all equivalent to each other; they can all be rearranged to get from one to the other, but which rearrangement is the most useful to calculate b? Just pause the video and think about that for a moment.

So, if we were gonna look to try and calculate b, we'd use the second one because b is the subject, and so we're gonna look at this a version a different part of the triangle is written as the subject.

So, it's helpful to use the version of the formula whose subject matches the part of the triangle that you're trying to find the value of.

In this example here we've got p, which is the opposite edge, it's opposite the angle 55 degrees.

We have the adjacent, which is 60 centimetres, and we have the angle theta of 55 degrees.

So, we're using the tangent formula because we have opposite and adjacent, and we know that the tangent formula involves both of those variables.

So, which of the three forms is the one where p is going to be the subject? Well, because p is the opposite, if we're trying to work out the opposite, then we're gonna use the form where the opposite is the subject, and then we can substitute our values in.

So, 60 multiplied by tan 55 degrees is equal to p, and we can use our calculator to calculate what that value is.

That value is 85.

69 to two decimal places.

It's worth noting here that when we are using our calculator, make sure that you are in degree mode.

Our angle is measured in degrees; that's our unit of our measurement, so we need to make sure our calculator is in that mode.

The quickest way to check that is to see if you've got a d on the top of your screen.

If it's an r or a g, then it's not in the correct unit, and you'll need to go into your settings and change that.

So, here's a check.

Which of these equations helps find the value of q most efficiently? So, look at the triangle.

Where is q, and which one is going to get you q in the most efficient way? Pause the video, and then, when you're ready to check, press play.

B is going to give you the value of q in the most efficient way because it is the subject.

No further rearrangement is necessary.

So hence, use your calculator and find the length of the side labelled q to two decimal places.

Pause the video whilst you calculate that, and then when you're ready to check your answer, press play.

64.

01 to two decimal places is the value of q.

So, we're into the first task of the lesson where you've got six right-angle triangles, and you need to use the tangent formula to find the length of the missing side that's labelled with a letter for each of them.

Round your answers to two decimal places.

So, you need to identify which one you are trying to calculate.

Is it the opposite or is it the adjacent? And then think about which form of the formula is going to be the most efficient.

Remember, it will be the most efficient if that part is the subject.

So, if you're trying to calculate the opposite, then use the form of the formula where the opposite is the subject.

If you're trying to calculate the adjacent, then use the form of the formula where the adjacent is the subject.

It might be that you need to go back in the video to find those forms and write those down.

But pause the video, and then when you've finished, we'll go through the answers.

So, the answers are on the screen.

On part A, you were calculating the opposite.

So, the opposite would have been 1 times tan 36 degrees.

1 because the adjacent has a length of 1.

So, 0.

73 is the answer to two decimal places.

On B, you are also calculating the opposite of this triangle.

So, you needed to do 10 times tan 36 degrees, and that came out as a value of 7.

27.

On C, you were also calculating the opposite.

So, you should have done 10 times 46 degrees.

And you can see here that C is now 10.

36.

It's now longer than the adjacent, and that's because 45 degrees would have been an isosceles triangle.

So, the two edges would have been equal in length, and now our angle has gone past 45 and is making its way towards 90, and so that length will get longer.

On D, you are now calculating the adjacent.

So, we needed to use a different form of the formula, and so we're going to use opposite divided by tan theta.

So, to work out D, you should have done 10 divided by tan 46 degrees, and that gave you 9.

66 to two decimal places.

On E, our angle 46 degrees is now in a different position to the previous four.

So, which one's the opposite and which one's the adjacent? Well, the opposite is the 46 centimetres, and the adjacent is the edge that we're trying to calculate, which is labelled as E.

So, you should have done 46 divided by tan of 46 degrees, and that gave you a value of 44.

42 to two decimal places.

And then finally, F.

F is the adjacent relative to the angle given, which is 10 degrees, and F is equal to 260.

88 centimetres to two decimal places.

Well done on that task.

So, we're now up to the second learning cycle, where we're gonna still be using the tangent ratio, and we're gonna be revisiting the tangent formula to find angles.

So, on the screen, we can see what's sometimes called a trig value table or a lookup table, and we've got the angles in multiples of five.

Then we've got the sine ratios, the cosine ratio, and the tangent ratios.

We are focusing on the tangent.

So, tangent is this ratio between opposite and adjacent, and we've been making use of that in the first learning cycle.

But now we're gonna look at, okay, if I know the opposite length, if I know the adjacent length, then how can I figure out what angle they are relative to? So, tan theta is equal to opposite over adjacent.

As we can use this example here that our opposite is 0.

577 units and our adjacent is one unit.

So, if I put the opposite value as the numerator and our adjacent value as the denominator, we are then asking ourselves this question: What is the size of angle theta given the ratio between the opposite and the adjacent is 0.

577 to one? Well, this is where we're gonna use our lookup table.

If we find in the tangent column that ratio of 0.

577, then that tells us that the angle is 30 degrees.

So, for any triangle, any similar triangle to this one, where the opposite and the adjacent are in this ratio of 0.

577 to one, then the angle would have been 30 degrees.

So, let's have a look at this one.

We have an opposite of 1.

4 units and an adjacent of two units.

So, 1.

4 divided by two, we can simplify that to 0.

7.

So, we're asking ourselves, what value of theta, when you apply the tangent function to it, gives you an output of 0.

7? So, head to the lookup table, look at the values.

In the tangent column, where do we find 0.

7? We find 0.

7 for an angle of 35 degrees.

So, the angle of 35 degrees gives you the ratio of 0.

7 to one between the opposite and the adjacent.

And again, any similar triangle where the opposite and the adjacent are in that ratio will mean that the angle theta is 35 degrees.

So, here is a check for you.

Which angle would give a value of 1.

732 for tan theta? Pause the video, and then when you're ready to check, press play.

So, you should have just had to find that in the column and trace that across to the angle column, and that tells us that this is 60 degrees.

So, a ratio between the opposite and adjacent of 1.

732 to one means that the angle would be 60 degrees.

So here, we've got the unit circle, or a sector of the unit circle.

It's the first quadrant.

The unit circle is a circle on our coordinate axes grid, where it's centred at the origin, so the point zero zero, with a radius of one.

On this particular diagram, along the circumference of the unit circle, you can see the angle in terms of degrees, and that angle is the rotation of the radius from the x-axis.

So, here I've got a line that's passing through the origin, passing through that point of 30 degrees up to the tangent to the circle.

And if we take the y-coordinate, so the height of the opposite, that's 0.

577.

And so, we can then trace that back to our table to see that that means that the angle of rotation would have been 30 degrees.

And we know it's 30 degrees from this diagram because of the points along the circumference, but that just shows you how the lookup table values, and this unit circle match up.

So, if we've got this triangle, 0.

5 is our opposite, it's opposite the angle marked theta, and 1.

2 is our adjacent length, it's between the angle theta and the right angle.

Then we can see that the tan theta has this ratio of 0.

417.

So, how do we make use of this on the unit circle? Well, if we trace across from 0.

17 on our vertical y-axis until it meets the tangent line, because this is tan, so we'd need that tangent, and then draw our line segment that starts at that point on the tangent, that's the point of intersection to our origin, so it's our extended radius, then we're looking to see what angle that has created.

Well, it's not passing through 20 or 25, so we're gonna have to estimate this a little bit.

And so, our angle theta is this angle of rotation, it's approximately 22.

5 degrees.

And so, on our lookup table, we wouldn't have found 22.

5 degrees, our lookup table of trig values only had multiples of 5 degrees, so this unit circle is allowing us to look at angles that are not just multiples of 5.

Here's another one, our opposite is one unit, and our adjacent is 1.

4 units.

So how do we use the unit circle? Well, we've got this value 0.

714, and that is our tan theta value, tan of theta is equal to 0.

714.

On the unit circle, with our tangent drawn at x equals 1, 0.

714 is the y-coordinate for the point at which a line passing through the origin intersects the tangent.

So, let's find that point, so we go across at 0.

714, because that's our y-coordinate, to our tangent, and then we can see that that's the point of intersection between a line that passes through the origin and our tangent.

And so, we can see that this is the angle theta, this is the angle of rotation that has got to that height.

And so, what is that using the scale along the circumference of the unit circle? Well, it's not 30 degrees, we've passed that, it's not quite 40 degrees, we haven't reached that yet.

It's gone slightly above 35 degrees, it's approximately 35.

5 degrees.

So, here's a check for you, the triangle has been evaluated to a decimal, we've mapped across to the tangent drawn on our line, so what is the approximate value of theta? Pause the video, and then, when you're ready to check, press play.

It's approximately 39.

8.

I'm hoping you haven't gone for 40 exactly, because it isn't passing through that dot in the centre, but it's very close to.

So, anything around 39, 39.

5, 39.

8, 39.

9, hopefully you're getting the gist, is a good approximation using that diagram.

So, the tangent function, which we write as tan of theta, returns the ratio of opposite and over adjacent for a given value of theta.

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

And your calculator means you can do this quickly and accurately.

So, in the same way that we were going to the lookup table, finding the value of our ratio, and then going to the angle, our calculator can do this for us, and therefore we can be much more accurate.

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

It does it in the opposite direction.

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

714, then we're going to use inverse tangent function to this equation to find that theta is the inverse tangent function of 0.

714.

We want to go from the ratio to the angle.

This has its own notation.

So, you will see it written with the inverse function notation as tan minus one.

It looks like a power of minus one, but it doesn't mean the same thing.

It means inverse.

Alternatively, its formal name for the inverse tangent function is arc tangent.

So, we shorten that to arc tan.

So, either of these are equivalent correct notation for using the inverse tangent function.

And this is programmed onto your calculator.

So, you can find this on your calculator using the shift or the second function key and then the tangent key.

And it will, the way that you'll see it on the screen, is using the inverse notation.

Check that you've got the D to mean that you're on the degrees mode.

And the answer there is your answer for your angle, theta.

And you can see how much more accurate this is than us using a table of values.

It's only gonna give you a multiple of five degrees, or the unit circle where we're estimating.

So, 35.

52683659 degrees with eight decimal places.

And we would clearly round that to some degree of accuracy.

So, 35.

5 to one decimal point.

So, onto the last task, question one, I want you to draw the appropriate lines onto the graph to estimate the angle in each of these two triangles.

So, pause the video whilst you're doing question one.

And then, when you press play, we'll move to question two.

Here's question two.

You need to calculate the missing angle in each triangle using your calculator and give your answer to one decimal place.

So, pause the video.

And then, when you're finished with those four parts, press play, and we'll go through the answers to all of task B.

So, task B, question one, you need to draw appropriate lines that included the tangent.

So, you needed to draw the tangent at x equals one.

And then you needed to calculate the ratio of opposite and adjacent.

So, seven divided by nine is 0.

7 recurring.

So that line goes across until you meet the tangent, and then draw the line that passes through the origin and that point.

Where that intersects the unit circle is where you will read off your approximation of the angle.

So, for A, it was 37.

9 degrees.

And these are approximate answers.

So, you will have something slightly different potentially.

On B, you need to do the same thing, work out the ratio of opposite and adjacent.

So opposite is 55, the adjacent is 60.

So opposite divided by adjacent, and that's 11 twelfths.

And then you needed to go across at that decimal equivalent until you hit the tangent, and then draw the line that passes through the origin up to that point.

It's an extended radius of the unit circle.

Where that line passes through the unit circle, where it intersects with the circumference, is where you'd read off your angle.

So, 42.

5 degrees or somewhere in that ballpark.

Question two, the more usual way of calculating the angle is using your calculator and using arc tangent.

So, 50.

0 degrees to one decimal place was part A.

55.

0 degrees was the answer to part B to one decimal place.

45.

0 degrees, the answer to C.

C, you didn't actually need to use arc tangent at all.

If you noticed it was an isosceles triangle with the given sides and it was a right-angled isosceles triangle, then you can calculate that that would be 45 degrees.

But because the answer was asked to be to one decimal place, then it's important that you've put 0.

0.

And then D, opposite is 5.

36 metres, adjacent is 20 metres, so you needed to do arc tangent of 5.

6 divided by 20.

And the answer is 15.

0 degrees to one decimal place.

To summarise today's lesson on checking and securing understanding of tangent ratio problems, the tangent ratio involves the opposite, the adjacent, and the angle.

The relationship between these can be expressed in three different ways.

We've got the formula where the opposite is the subject, the formula where the adjacent is the subject, and the formula where tan of the angle is the subject.

This tangent formula can be used to find the length of the opposite, the length of the side adjacent to the angle theta, or the size of the angle theta itself.

In order to find the size of the angle, the function inverse to the tangent function must be used, and that's known as the arc tangent function.

On a calculator, the arc tangent function is usually written as tan minus one, and that's because it's the inverse function notation.

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