### Video Transcript

The diagram shows a roundabout in a playground. Johnny is sitting at point ๐ต, and his mum is standing behind a fence at point
๐ด. The fence is perpendicular to the line ๐ด๐ต. Johnny moves anticlockwise to point ๐ถ, which is a perpendicular distance of ๐ฟ from
the fence. Angle ๐ถ๐๐ท is equal to ๐ฅ degrees.

Part a) Show that ๐ฟ is equal to six plus three sin of ๐ฅ degrees metres.

Itโs not instantly obvious how to answer this problem, but we can break it down into
smaller steps. The first step is to find the horizontal distance of the point ๐ท from the fence. Then we find the distance between the points ๐ท and ๐ถ. And adding those two values will give us the length ๐ฟ.

Now we know that the line joining the centre of the circle to its circumference is
the radius. So the radius of our circle or our roundabout is three metres. And therefore, we can add in another radius in this circle, which must also be three
metres in length. We can see that the roundabout is three metres away from the fence. So the horizontal distance of point ๐ท to the fence can be found by adding three
metres and three metres. Thatโs six metres.

Now we just need to find the length of the line ๐ท๐ถ. We can see that the line ๐ท๐ถ makes out part of a right-angled triangle, with an
included angle of ๐ฅ degrees. The hypotenuse of this triangle โ remember, thatโs the longest side and the one that
sits opposite the right angle โ is the radius of the circle, so it must be three
metres. And we can use right angle trigonometry to find an expression for the line ๐ท๐ถ in
terms of ๐ฅ.

๐๐ถ we said is the hypotenuse of the triangle, and ๐ท๐ถ is the opposite. Thatโs the side opposite the included angle. The remaining side is the adjacent. Thatโs the one next to the included angle. In fact, we donโt need that side.

We know the length of the hypotenuse, and weโre trying to find the length of the
opposite side. We use the sine ratio. Sin ๐ is opposite over hypotenuse. Substituting what we know about our triangle into this formula gives us sin of ๐ฅ
degrees is equal to ๐ท๐ถ over three.

We can solve this to form an expression for ๐ท๐ถ in terms of ๐ฅ by multiplying both
sides by three. And that tells us that ๐ท๐ถ is equal to three sin of ๐ฅ degrees. We said that the length of ๐ฟ was equal to the distance of ๐ท from the fence, which
we said was six metres, plus the distance ๐ท๐ถ. Thatโs six plus three sin ๐ฅ. Since our measurements are in metres, ๐ฟ is equal to six plus three sin of ๐ฅ degrees
metres, as required.

Now at this point, we do need to clear a little bit of space to move on to part
b. However, since this is a show that question is hugely important that you leave
everything youโve written down on your paper.

Part b) Johnny continues to spin anticlockwise around the roundabout until he is at
point ๐. Work out the length ๐ฟ now.

We can use the formula we just created, this time with an angle of 130 degrees. Substituting 130 degrees into our formula gives us six plus three sin of 130, which
is 8.2981 metres.

Now no level of accuracy is specified in the question. However, since this is in metres, if we round to the nearest one hundredth, that will
be like rounding to the nearest centimetre. The deciding digit, eight, is greater than five, so that tells us to round the nine
up. However, when we round the nine up to a 10, we canโt fit the digits one and zero in
this column, so we carry the one and two becomes three. 8.2981 rounds to 8.30 metres. And the length ๐ฟ now is 8.30 metres.

Part c) State the size of angle ๐ฅ for which the length ๐ฟ is smallest and state the
value of ๐ฟ.

Letโs consider the graph of ๐ฆ is equal to sin ๐ฅ. Itโs a periodic function; that is, it repeats, and has a period of 360 degrees. Its maximum is when ๐ฅ is 90 degrees, and that gives us one. And its minimum is when ๐ฅ is equal to 270 degrees. That gives us a value of negative one.

The smallest value in our function then must occur when ๐ฅ is equal to 270. We donโt actually need to include the degree symbol here since itโs included in the
formula. We know at this point that sin of ๐ฅ is equal to negative one. So we can substitute negative one into our formula for ๐ฟ. And that gives us six plus three multiplied by negative one, which is three. The size of angle ๐ฅ for which the length ๐ฟ is smallest is 270, and the value of ๐ฟ
here is three.