Measuring Angles

Transcript

0:00Introduction

Hello everyone! Let us begin with a small puzzle. The minute hand of a wall clock is seven centimetres long. In twenty minutes, it turns through one hundred and twenty degrees. So how far does its tip travel? Arc length is radius times angle, so seven times one hundred and twenty, which gives eight hundred and forty centimetres, right? Hmm, more than eight metres, from the tip of a small clock hand? Something is badly wrong here. Today we will see how angles are really measured, and exactly where this answer went wrong. Questions on radians, arcs and sectors come up often in JEE, so let's begin.

0:40What is an angle?

So, first things first, what is an angle? An angle is the amount of rotation of a ray about its starting point. Okay? And the direction of turning matters. Anticlockwise rotation gives a positive angle, and clockwise rotation gives a negative angle. Right? Now let me ask you this. Can an angle be more than one full turn of the ray? Yes, of course it can, because the ray may turn through more than one full revolution. So an angle can have any real value, positive or negative, as large as you like. Clear?

1:15Unit one: the degree

Now, to measure this rotation, our first unit is the degree. One degree, written with the small raised circle, is one over three hundred and sixty of a full revolution. So one full turn is three hundred and sixty degrees. Got it? One degree equals sixty minutes, and a minute is written with a single small dash, called a prime. And one minute equals sixty seconds, written with two dashes, a double prime. With me?

1:42Unit two: the radian

Now the second unit, and for us the more important one, the radian. Draw a circle, and mark an arc whose length is exactly equal to the radius. The angle this arc makes at the centre is one radian. Okay? So, tell me, how many radians make one full revolution? The answer is two pi radians, where pi is the Greek letter you know from the circle, and rad is short for radian. Right? Now, a full revolution is also three hundred and sixty degrees. Divide both by two, and we get the one line to remember: pi radians equals one hundred and eighty degrees. Clear? Divide both sides by pi, and one radian equals one hundred and eighty degrees over pi. The wavy equals sign means approximately, so one radian is about fifty seven degrees, seventeen minutes, forty five seconds. Got it? And the other way round, one degree equals pi over one hundred and eighty radians, which is about zero point zero one seven four five radians. With me?

2:42Converting, and the missing unit

So converting is really simple. To go from degrees to radians, multiply by pi over one hundred and eighty; to go back from radians to degrees, multiply by one hundred and eighty over pi. The arrow means converted to, and the cross means times. Okay? Now, suppose a question writes sine two, with no unit at all. Is that the sine of two degrees? No, and this is a common slip. When no unit is written, the angle is in radians, so sine two is the sine of two radians, not of two degrees. Clear?

3:15Arc length and sector area

Now see why radians are so useful. On a circle of radius r, an arc of length r makes one radian at the centre. So if the arc has length s, what angle does it make? Yes, exactly! It makes s over r radians. Theta, the Greek letter, is the angle, so theta equals s over r, and s equals r times theta: arc equals radius times angle. Okay? Next, the area of a sector, the slice of the circle between two radii. Its area, capital A, equals one half times r squared times theta, which is the same as one half times r times s. Right? And now the most important warning of this video. Both these formulas are true only when theta is in radians! If the angle is in degrees, convert it to radians first. Our clock puzzle broke exactly this rule, and we will fix it in a minute. With me?

4:08Clock hands and polygons

First, clocks. Let capital H be the hours and capital M the minutes; on screen, small h means hours and min means minutes. The angle between the two hands is the mod of thirty times H, minus eleven over two times M, in degrees. The two vertical bars are the mod sign, so the value is never negative. Got it? And if this number is more than one hundred and eighty degrees, take three hundred and sixty degrees minus it. Okay? Now polygons. In any n-gon, a polygon with n sides, the interior angles add up to n minus two, times pi. And in a regular n-gon, all the angles are equal, so each interior angle is n minus two, times pi, over n. Right?

4:51Example 1: distance moved by a 7 cm minute hand in 20 min

Now, let's settle our clock puzzle properly. The minute hand of a clock is seven centimetres long. How far does its tip move in twenty minutes? Take pi as twenty two over seven. Pause the video and try it first. Have you got an answer? Okay, let's check. The minute hand makes one full turn, two pi radians, in sixty minutes. So in twenty minutes it turns twenty over sixty, times two pi, which is two pi over three radians. Okay? Now the angle is in radians, so s equals r times theta is safe to use. Seven times two pi over three gives fourteen pi over three. Right? Now put pi as twenty two over seven. Fourteen over three, times twenty two over seven: the seven goes into fourteen twice, so we get forty four over three centimetres. That is about fourteen point six seven centimetres, a sensible answer, not eight hundred and forty! The mistake in our puzzle was putting one hundred and twenty degrees straight into s equals r times theta. Got it?

5:54Example 2: greatest area of a sector with perimeter 20 cm

Now a harder one. Here, a sector of a circle has perimeter twenty centimetres. Find the radius and the angle for which its area is greatest, and the greatest area. Pause the video and try it first. Here is a hint for you: start with the perimeter. Have you tried it? Okay, let's check together. First, the perimeter. The perimeter of a sector is just its arc, r times theta, right? No, no, wait, a sector also has two straight edges, the two radii! So the perimeter is two r plus r times theta, and that equals twenty. That gives r times theta equals twenty minus two r. With me? Now the area. Substituting, one half times r squared times theta becomes one half times r, times twenty minus two r in brackets, which is ten r minus r squared. Okay? Now we want the greatest value. Write ten r minus r squared as twenty five, minus r minus five whole squared. Since a square is never negative, the area is greatest when r equals five centimetres. Right? Then theta equals twenty minus ten, all over five, which gives two radians. And the greatest area is ten times five, minus five squared, so fifty minus twenty five, that is twenty five square centimetres; cm with a small two up top means square centimetres. If you spotted the two radii in the perimeter, very good, that is exactly the trap. Clear?

7:26Quick revision: radians, arc, sector, clocks, polygons

So, quick revision. Pi radians equals one hundred and eighty degrees, and when no unit is written, the angle is in radians. Arc length, s, equals r times theta, and sector area is one half times r squared times theta, with theta in radians only. And for clocks, use the modulus of thirty times H, minus eleven over two times M, while any n-gon has angle sum n minus two, times pi. Now go and try the practice problems on this page. All the best!