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 are looking forward to learning something new.

I'll be there to support you as we go through the lesson.

So today's learning outcome is to be able to get the trigonometric ratios for 30 degrees and 60 degree angles.

We've got a few keywords that we'll be using during the lesson.

None of them are new to you, but we will take the time now to familiarise ourselves before we start the lesson.

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

The next few definitions are relative to the diagram that you can see on the screen.

So there is a unit circle on that diagram, which is the circle centred at the origin with a radius of one.

And then we can see right-angled triangles where we've got a radius of one as a hypotenuse, but also the radius of one is one of the edge lengths of a right angle triangle.

So the sine of an angle is the Y coordinate of point P on the triangle formed inside the unit circle.

So point P is along the circumference of the unit circle and the hypotenuse is the radius with a length of one.

The cosine of an angle is the X coordinate of the point P on the triangle formed inside the unit circle.

So once again, that point P is the same point P that was defined in the sine definition, but this time cosine is the X coordinate.

A tangent to a circle is a line that's intersects the circle exactly once.

And then we get to our third trigonometric ratio, which is the tangent of an angle is the Y coordinate of point Q on the triangle, which extends from the unit circle.

So if you look again at the diagram, we've got point P and point Q.

You may wish to pause the video so they can read through them at your own speed and check alongside the diagram.

Press play when you're ready to move on.

So this lesson on calculating trigonometric ratios of 30 degrees and 60 degrees is split into two learning cycles.

In the first learning cycle we are going to find the exact values for 30 degrees and 60 degrees.

And then later on when we get to the second learning cycle, we'll be spotting relationships making use of the 30 degrees and the 60 degrees.

So let's go ahead now and derive the exact values for 30 degrees and 60 degrees with our trigonometric ratios.

So when working with trigonometry, we often only see rounded values and you can see some of those on the calculator displays.

And this is because the calculator can only display a certain number of digits.

So in this case here we can see 10 decimal places.

The left hand calculator display is showing the value of sine of 38 degrees.

I know it's degrees because there is the capital D on the top line of the display that indicates what units my angles are, and that in this case is degrees.

The right-hand side is showing us the value of cosine of 12 degrees.

So we need to see that they are rounded values.

Certain trigonometric ratios can be expressed exactly.

Jacob's saying, "Which ones and how can I tell?" So first of all, what do we mean by exactly? Pause the video and think about other parts of your maths knowledge, the way you've used exact values.

Press play and we'll discuss that a little bit further.

So places where you'll have used exact values will have been working with surds, when working with pi.

So values that are not rounded.

As soon as you put some sort of degree of accuracy, so maybe you've truncated the value, maybe you've rounded the value, maybe you've rounded it to decimal places or rounded it to significant figures, you've now included a rounding error.

It's an approximation of the answer to a degree of accuracy.

Whereas if we work in terms of pi or work with surds, then these are exact forms, no rounding has been included.

So which trigonometric ratios can be expressed exactly? Well, it's 30 degrees and 60 degrees that we can, there are some others as well.

But this lesson we're looking at how we can derive and find the exact values of 30 degrees and 60 degrees.

So let's start by considering an equilateral triangle.

So on the screen there is a diagram of a triangle that is an equilateral triangle.

Pause the video and think about what other information we know if this is an equilateral triangle.

Press play when you're ready to move on.

So I'm hoping you thought about the fact that an equilateral triangle, all edges are equal.

So the other side would be two units, and the one unit is because it is half of the side length.

How do we know it's half of the side length? Well, because that perpendicular line is the altitude, but it's also the line of symmetry.

And that altitude and that line of symmetry are perpendicular by definition and also bisects the edge that it meets.

And that's why that is half.

What else do we know about equilateral triangles? equilateral triangles, all the edges are the equal, but also all the angles, the interior angles are all the same.

So here is a check thinking about those properties of an equilateral.

What is the size of the angle marked A and what is the size of the angle marked B? So pause the video and when you're ready to check your answers for those two questions press play.

So the first one, A, is an interior angle of the equilateral triangle and all angles inside of any triangle sum to 180 degrees.

But being an equilateral triangle means that all three angles are equal.

So we can divide 180 by three and that gives us 60 degrees.

So what did you get from angle marked B? Well, the angle marked B is 30 degrees and that's because we've got that line of symmetry, that altitude that is bisecting the angle.

So half of 60 degrees is 30.

So Pythagoras's theorem can then be used to calculate the remaining length.

So we've started with an equilateral triangle and we've actually only working now with half of it.

We split it into two congruent right angle triangles.

So what is the missing edge length? Pause the video and try and calculate that, reminding yourself about Pythagoras's theorem and then when you press play we'll go through what Pythagoras's theorem is and what the side length is for this triangle.

So Pythagoras's theorem only holds true for a right angle triangle, but that's okay, we have a right angle triangle here.

It also states that the sum of the square of the two shorter sides is equal to the square of the hypotenuse.

and that's how we're gonna calculate the missing side length.

Our hypotenuse is two, it was one of the edges of the equilateral triangle.

We know it's the hypotenuse because it's the longest edge, but also because it's opposite the largest angle, the right angle.

So we're going to find the square of that, which is four.

And the square of the shorter side that we know, which is one.

And the difference between them will be the square of the unknown side.

If we put the square of the unknown side, which is three, four minus one, then we're gonna square root it to get the side length.

We don't want to round that value, we're trying to work with exact values.

And as we've already said, a surd is the exact form.

So we're gonna leave that third edge in terms of square root three.

So we've got hypotheses of two, a shorter side of one, and the other short side is root three units.

And now we've got all three angles inside the triangle and all three edge lengths, all as exact values.

We can now deduce the exact values for the trigonometric ratios for either the 30 degree angle or the 60 degree angle.

So sine of 60 degrees is root three over two, and this is the exact value for sine of 60 degrees.

Where did the root three over two come from? Well, sine is the ratio between the opposite and the hypotenuse.

So if we look at where the 60 degree angle is, then the opposite side is root three in its length.

The hypotenuse is opposite the right angle and that has a length of two.

So sine of 60 degrees is root three over two.

So here's a check for you, what would the exact value be for sine of 30 degrees? Pause the video and then when you're ready to check press play.

So sine of 30 degrees would be equal to a half, one over two.

And that's because where is the opposite to the 30 degree angle? Well, that would be the one unit length.

The hypotenuse is still the two because it's opposite the right angle.

So opposite over hypotenuse one over two.

So the exact trigonometric value for sine 30 degrees is a half.

So we're up to the first task of the lesson.

There's only one question in this task and you need to deduce the exact values for the cosine and tangent of both 30 degrees and 60 degrees.

So in a very similar way to how we just did sine of 60 degrees and sine of 30 degrees, you now need to do that for cosine and tangent.

The hint is to construct an equilateral triangle of length two units.

And so you'll be working with a very similar triangle to how we deduced the sine values.

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

So you needed half of the equilateral triangle and the reason we need half is so that we can have that 30 degree angle as well.

We use Pythagoras's theorem in the same way to get our third edge, which is root three.

And now we can use the definition of each ratio to come up with the exact trig value.

So for cosine, cosine is the ratio between the adjacent and the hypotenuse.

So if it's cosine of 30 degrees, the adjacent is root three.

The hypotenuse is two, so the exact trigonometric value is root three over two.

Cosine of 60 degrees, so once again, we're gonna look for the adjacent to the 60 degrees and that is the one unit edge.

The hypotenuse is two, so cosine of 60 degrees is equal to a half.

Then if we move to the tangent, well tangent is the ratio of the opposite and the adjacent.

So for tan of 30 degrees, the opposite is one, the adjacent is root three.

So one over root three, but that is equivalent to root three over three.

And remember, we should always rationalise the denominator where possible.

So tan of 30 degrees is equivalent to root three over three.

And lastly, tan of 60.

Well again, where's the opposite? Well, the opposite is root three.

Where's the adjacent to the 60 degree angle? Well, that's the one.

So root over one is just equal to root three.

So cosine 30 is equal to root three over two.

Cosine 60 is equal to one half.

Tan of 30 or tangent of 30 degrees is root three over three and tangent of 60 degrees is root three.

So we're now up to the second learning cycle where we're gonna spot relationships and make use of these exact trig values for 30 degrees and 60 degrees.

So firstly, do these exact values only hold if the right angle triangle has sides of these exact lengths? We deduced them using this triangle, so is it only this triangle that they are true for? Pause the video and think about that a little bit before we move on.

Think about what the trigonometric ratios actually mean.

Press play when you're ready to look at this a little bit further.

So the answer is no.

And the knowledge of similar triangles tells us this.

Trigonometric ratios are a ratio between two sides of a right angle triangle, regardless of its size.

We can use the unit circle to help us with this concept of finding the value and then we can scale it up for any size triangle.

And that's exactly the same here.

We can make use of similar triangles.

So any right angle triangle that has got an angle of 60 degrees and an angle of 30 degrees will be similar to this one here.

And so the ratio between the sides, the two sides, depending on which trigonometric ratio you're using, will always simplify back down to root three over two or one over two.

Or any of the other trigonometric exact values.

So if we've got these two right angle triangles, they are similar, we know they are similar by the angles.

So there's a 90 degrees, there's a 60 degrees, which infers that the other angle is 30 degrees.

One has got the adjacent as one, and the hypotenuse is two, the other one has got the adjacent as 10.

And the hypotenuse is unknown.

But we can make use of similar triangles and thinking about a ratio table to work out X quite easily.

So if we look at the adjacent, the adjacent is 10 times as long if the unit is centimetres, than the equilateral triangle that we deduce the exact trig values from.

And so this ratio between the two sides will be the same on the hypotenuse.

And so X is therefore 20.

So that's one way of us seeing that X is 20.

By making use of the proportions and the multiplicative relationship between the two triangles.

But how can we make use of the exact trig values that we've just looked at? Well, we know from the left hand triangle that the exact trig value for cosine 60 degrees is equal to a half.

And therefore 10 over X is also equal to a half.

And so if we write that down as an equation, cosine 60 is equal to a half.

We know that because it's an exact trig value, but our triangle is not one over two but instead 10 over x.

And so by solving that equation we can see that X must be 20.

What fraction where the numerator is 10 is equivalent to a half? What denominator does that need to be? So here's a quick check.

True or false? Exact values only happen when the right angle triangle has side lengths of one, the root three, and two? Pause the video and justify why you've gone for true or false depending on what you've chosen and then press play when you're ready to check.

So it's false and hopefully you saw that in the example with the similar triangles.

So similar triangles have a multiplicative relationship, therefore any triangle similar to one with a stated side length will produce exact values.

They will just simplify down to root three over two or a half.

So what the values of X and Y in the triangle below.

So let's have a look at what we know.

Well, we know that the opposite to the 30 degree angle is three.

We haven't got the adjacent, that's something we need to work out, and we haven't got the hypotenuse.

That's something we also need to work out.

But what we do know is that there are some exact trigonometric values for 30 degrees and two of them are on the screen.

So tan of 30 degrees is equal to root three over three and sine of 30 degrees is equal to a half.

So for any similar triangle, so for any triangle that has a 90 degree and a 30 degree angle, that ratio between opposite and adjacent and opposite and hypotenuse is exactly the same.

So a half, our sine of 30 degrees, which is our opposite over hypotenuse, for our question here, the opposite is three, and the hypotenuse is Y.

So sine of 30 degrees is also equal to three over Y, opposite over hypotenuse.

So by creating this equation, equating the exact trigonometric value to our ratio, we can then solve it.

Think about how you would solve that equation.

What's the value of Y? Well, the value of Y is six.

But let's just step back a moment.

You'd have been able to answer that question before this lesson.

Even if you didn't know that sine of 30 degrees is equal to a half, you'd have been able to use the sine formula to solve and find Y.

And how would you have done that? You'd have done three divided by sine 30.

And on a calculator it's gonna give you the answer of six as well.

Three divided by a half is equal to six.

So let's now look at trying to work out X.

So X is the adjacent on this triangle to the 30 degree angle and we have got the opposite, which is three.

So we can make use of the exact trig value tan of 30 degrees.

So tan of 30 degrees is root three over three.

And so tan of 30 degrees on this particular triangle is three over x, the opposite over the adjacent.

So we can equate the exact trig value to our ratio of opposite and adjacent.

And once again, we now just need to go through our algebraic skills to solve this, to calculate what X must be.

And X would equal three root three, in its exact form.

And we're going to be using this without a calculator in the most part.

But if we did have a calculator, if we didn't remember or know that tan of 30 degrees has an exact trigonometric value, we would still be able to calculate X.

We'd use our tangent formula to calculate the adjacent.

And we would do that by doing opposite divided by tangent theta.

Three over tan of theta is three root three.

So Laura said, "But why do I need to know these?" So all of these exact trigonometric values for 30 degrees and 60 degrees, what's the use of them? Why do I need to know them? Can't I just use my calculator? Perhaps you've been having that thought as you've been going through the lesson.

Sam said, "This is a useful way of checking you have the right answer, and it also means you don't have to rely on your calculator." So up to the final task of the lesson and question one, you need to find the lengths of the unknown sides without using a calculator.

And you'll be able to do that because they are all including 30 degree angles or 60 degree angles, which we now know have exact trigonometric values.

Pause the video and when you're ready for the next question press play.

So here's question two, and Lucas's calculator has broken, so he cannot rely on the calculator.

Explain how he can use his knowledge of similar triangles to complete his homework.

What are the exact values of X and Y in the triangle below? So pause the video and calculate the exact values of X and Y and how we can make use of similar triangles to answer his homework.

Press pause and then when you're ready for the answers to task B, press play.

So on question one we're going through the answers.

So part A, it was exact trigonometric value of 30 degrees.

It was the opposite that you were trying to calculate given the adjacent.

So you should have been using tan of 30 degrees.

And A would be equal to root three over three.

On B, it was 30 degrees as well.

It was also the opposite you were trying to calculate and you were given the adjacent.

So this is actually a similar triangle to part A.

The adjacent is 10 times as long and therefore the opposite to continue with the same ratio will also be 10 times as long.

So B is equal to 10 root three over three.

On C, we are looking to work out the opposite once again.

We have the adjacent, but this time the angle has changed.

So the angle is 60 degrees.

Tangent of 60 degrees is root three.

So root three is equal to C over 10 if we equate those two things, which means that C is equal to 10 root three.

Onto D, D is the hypotenuse, 10 is the opposite.

So opposite and hypotenuse are in the sine ratio.

Sine of 60 degrees is root three over two.

So you needed to say that root three over two is equal to 10 over D, and then solve that to find D.

D is therefore 20 root three over three.

You can see the rationalised form and also the the form where the denominator is irrational.

Onto E, we've got a 60 degree angle once again.

E is the adjacent to that 60 degree angle, and 46 is the hypotenuse.

So you should have been using cosine 60 degrees.

Cosine of 60 degrees is a half.

So one half is equal to E over 46, which means the E is 23.

And finally F, F is the hypotenuse, 46 is the opposite.

The angle is 30 degrees, sine of 30 degrees is equal to a half, and therefore F is 92.

Question two, Lucas's calculator was broken so we need to make use of similar triangles and exact trigonometric values in order to get the value for X and Y.

So tan of 60 degrees is equal to X over eight.

The opposite over the adjacent.

Tan of 60 degrees has an exact value of root three.

So if X over eight has to also equal root three, then X would have to be eight root three.

Then if we're trying to calculate Y, we can make use of the eight as the adjacent to the 60 degree angle.

And Y as the hypotenuse means that cosine 60 degrees is equal to a half.

So eight over Y, the adjacent divided by the hypotenuse, is also equal to a half.

And we can solve that to say that Y is equal to 16.

You could have calculated X or Y, whichever one you calculated second by using Pythagoras's theorem instead.

So to summarise today's lesson, trigonometric ratios for 30 degrees and 60 degrees can be expressed as exact values.

And they are derived from an equilateral triangle with side lengths of two units.

Although you don't have to use an equilateral triangle with side lengths of two units, any equilateral triangle would work because it is similar and then they would simplify down to the exact trig values that we've learned during the lesson.

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