Angles (1978)

[Music] ready [Music] hi Island do you think it would be sensible to just sit here admiring the scenery while they cut down this tree well I don't know if it would be sensible it would be a bit wiser though if you could actually calculate the height of the tree and you can actually do that if you know a bit about angles now let me illustrate with this carer's rule that I always carry about with me as you can guess intuitively we'd imagine that this is an angle it's a small angle this is a bigger angle what about this in fact this is the same angle as I had previously what about this same angle again you see it doesn't depend on the legs what we need is some way of measuring angle in order to be able to do calculations using them and we're going to measure angle by using movement in a circle if it's a circle of radius r then the circumference is given by 2 pi r well actually it's clear that if you want to measure angles all you need is a horse on a lung Ren but actually it's clear that the horse is tracing out an increasing angle and what we'd like to do is to start off by measuring one of these angles so let's measure the angle that the horse has stopped at that's the angle that the rain has made and now I knew this meter stick would come in useful so let me Pace out meters and I've got 1 2 3 four so in fact the horse is traced out an angle which is measured on the circumference by 4 M well let's try measuring that angle again I suppose the horse is getting a bit dizzy so let's increase the radius yes well what about the angle now in fact I can see it's exactly the same angle as before so let me measure it by pacing around the circumference well I use my meter stick and take meter paces and I have 1 2 3 4 5 6 7 8 well that's very funny I have exactly the same angle but now I need 8 m to go around the circumference and last time I had four but that must be because we've changed the radius let's see what happened before well in fact the present radius is measured by 1 2 3 4 5 six it's 6 M and the old one I can see is 1 2 3 3 m so in fact all our measurements have changed and yet the angle has remained exactly the same well remember when we doubled the radius we actually doubled The Arc Length traveled so the ratio of Arc Length walked by the horse over radius Remains the Same so that must be the basis for our definition of the measure of an angle so let me write that down so when the horse traced out that angle the first time it walked 4 M the second time it walked 8 m but remember the first time the radius was 3 m and the second time the radius of the lerine was 6 M and it's that ratio which is the same each time so it must be that number 1.33 which gives us a measure of the angle traced out by the horse let me go over that again this angle has an arc 4 M long for a circle of radius 3 m so for this angle the ratio of the arc length over the length of the radius is 4 over 3 which is just 1.33 in decimal now we can make the radius longer and a large the circle but keep the angle the same even though both arms are longer what happens to the ratio well the arc length is now 8 m and the radius measures 6 M our ratio 8 / 6 is still 1.33 and for the same angle we get the same number whatever the size of the circle each angle has its own particular ratio which stays constant whatever size circle you look at the number we get as the ratio is a measure of the angle in radians well what about some other angles for example this angle is 0.5 radians the ratio of Arc to radius is a half here the angle is just one radian The Arc is the same length as the radius this angle is two radians at this point the angle is 3 radians and so on for a right angle we can calculate the number of radians with radius R the whole circumference is given by 2 pi r and a quar of that is the Arc of the right angle well that's just a half p r so so the ratio of Ark over radius half pi r / R is just a half pi so in radians a right angle is just Pi / 2 well we know that Pi is approximately 3.14 so in decimals half of Pi is roughly 1.57 a right angle is just 1.57 radians or Pi / 2 radians so the number for a right angle in this radian measure is 1.57 well you might be more familiar with 90° you know you measured that on this protractor when you were at school well in fact that comes from the Babylonian measure they took a circle and they divided that into 360° so a quarter of the way round would be 90 so 90° of course is another expression for a right angle and in fact degrees that's another way for measuring angles right now let's see that's 25 25 M 50 all right so that's the distance from the bottom of the tree to the bench 25 50 now of course it's all very well the story of Allen with his horse and his angles and his arcs and all the rest of but what I really want to know something something quite simple is the height of that tree there more or less than this distance I just measured now I wonder what Alan can tell us about that well trigonometry is the branch of mathematics that connects angles with Heights and distances look at this new angle it's 0.68 radians the ratio between the curved Arc and the radius is 0.68 but to deal with the case where the side that's opposite the angle is not an arc but an upright straight line we must look at ratios in a right angled triangle the side opposite the angle measures eight in some units of length and the base of this triangle is now 10 units so the ratio of the opposite side over the base is 8 over 10 0.80 well the upright could always have been drawn in another place to make a different size triangle if the opposite side now measures four the base is five the ratio is now 4 over 5 which is again 0.80 it turns out that whatever size triangle we take the ratio of these two sides always stays the same however this ratio of8 is different from the arc over radius ratio which was 0.68 radians so we must give it another name well if you take a protractor you can actually measure this angle you'll find it's 38.7 de and we call the8 ratio the tan of the angle that's just short for a tangent so we can write the tan of 38.7 de is 0.80 and tangents have been tabulated for every angle for example tan of 26.6 De is 0.5 the tangent of 45° is precisely one because the height and the base are exactly the same length and the tangent of 63.4 De turns out to be two in fact the tangent of any angle can easily be found all you've got to do is push the right button on your calculator well now we begin to see daylight you see because with this tangent idea uh that you've just seen explained so nicely by Alan I think I can solve the problem of the tree you see just think about it I'm already know the distance along the ground between the bench here and the bottom of the tree so if I can find some way of measuring the angle between the ground and the line from the bottom of the bench to the top of the tree that angle there then I can use the idea of the tangent of the angle to calculate the height of the tree so the problem is how do I measure the angle well I'm going to do it with this which I don't habitually carry around in my pocket but somebody kindly provided me with it Christopher Columbus would probably recognize a thing like that except for the plastic so what I have to do is to put that down by the edge of the bench and then just sight up this kind of gunsight thing here up to the top of the tree well you probably realized that the angle that really interests me is 45° because you remember alen explained that the tangent of 45° is one so with 45 degre you'd have a height and a base which are exactly equal and that would be the point at which I would be thinking about moving a little bit further back so let's see oh no that can't be right no I've gone a bit too high well that's just as well isn't it so say about say about there what have we got here um 40 43° all right I'll just see if I can work that out now well I suppose I call the height of the tree here H and then the angle I just measured is this angle here and that was 43° so I know that the tangent tan 43° is equal to the height of the tree divided by that length I measured 25.50 M so the height of the tree is equal to 25.5 time tangent of 43° so let's just work that out 43 tan time 25.5 equals well that's 23.7 20 wow 23.7 M so I've got about 2 m in hand right so let's go and Mark that off shall we two MERS well one two say about there so that's where I figure the top of the tree will come to and I've got a little margin of safety so maybe I can just go and sit and watch them cut it down [Music] [Music] ha