Memorizing Trigonometric Identities with Hexagon

What's a super hexagon? Does it really help us remember all the trigonometric formulae and identities we have seen so far? Yes it does. It's a simple hexagon which looks like this. All we need to do is join all the opposite vertices and that gives us three diagonals. We write the number one in the center. It's still difficult to believe that one hexagon can lead us to so many formulae. To know the formulae, we give the six function names to each of the six vertices. But there is a particular way in which we need to write it. We start writing from this vertex, go around the hexagon and end at this one. It's simple! We just draw a hexagon join its opposite vertices and write a 1 in the center. Now let's see, how we write the six functions at the six vertices. We just need to remember one formula which we have already seen. tan is equal to sine by cos. We have used this formula like a zillion times now. tan is equal to sine by cos. We start at this vertex and write tan, sine, cos in the clockwise direction. tan, sine, cos. Now three functions and three vertices remain. We write cot opposite to tan. Remember co-tangent opposite to tangent. What remains is cosec and sec. If we draw a vertical line through the center of the hexagon, we will have three vertices on the left and three on the right. So to remember what's written on this vertex just remember that we have all functions starting with c on the right side. So we write a cosec here and what remains is sec. That's all you need to remember about how the super-hex has to be drawn. Three diagonals joining the opposite vertices, the number one in the centre and the six function at the vertices. So let's draw the super-hex on a new page. A hexagon with three diagonals joining the opposite vertices. The number one at the centre, tan is equal to sign by cos, cot opposite to tan cosec here since all c's are on the right and what remains is sec. Are you ready? You're going to learn a whole set of trigonometric formulae. This video is fun but also very long as we will be covering most of the trigonometric formulae we have studied before. To understand the first set of formulae let's try going around the hexagon in the clockwise fashion. We go like this. This tells us that tan theta equals sine theta by cos theta. Well that's how we started making our hexagon. Now lets see the next three functions in clockwise direction. These three! This tells us that sine theta equals cos theta by cot theta The first one equals the second one by the third one. Take the next three functions clockwise. First equals second by third. Hence cos theta equals cot theta by cosec theta. And if we take the next three functions, we can say that cot theta equals cosec theta by sec theta. And we will get two more formulae if we go clockwise. Cosec is equal to sec by tan. And sec is equal to tan by sine. Wasn't that simple? One hexagon and six formulae without any effort. Great! But what if you forget that you had to go clockwise and you go anti-clockwise instead. Let's say you started like this, in the anti clockwise direction. Don't worry! The pattern still holds true. First will equal second by third. tan theta equals sec theta by cosec theta. First equals second by third. If we take these three functions anti-clockwise, we can write sine theta equals tan theta by sec theta. If these three, then cos theta equals sine theta by tan theta and the list goes on. Whether you go anti-clockwise or clockwise, first will always equal second by third. We can easily get three more here. One simple hexagon and twelve formulae. Is that all? No, there's more! Let's draw the super-hex on a new page to understand what else there is! We draw the super-hex and write a 1 in the centre. tan is equal to sine by cos, cot opposite to tan, as we have all c's on the right, we write a cosec here and what remains is sec. In the previous set of formulae we did not use the one which was written in the centre. Let's see why it is placed like this. If we multiply the functions at the opposite vertices, we get a 1. This multiplied by this equals 1. sine theta multiplied by cosec theta equals 1. This multiplied by this also gives us 1. cos theta multiplied by sec theta equals 1. And this multiplied by this also equals 1. tan theta multiplied by cot theta equals 1. So when functions are diagonally opposite vertices are multiplied, we get a 1. Wait, there's more. We draw the super-hex, right the number one at the center. tan equals sine by cos, cot opposite to tan and cosec here since all c's have to be on the right and what remains is sec. I repeat the positions again and again so it gets etched in your brain after watching the video. Take any three continuous functions in the hexagon like tan, sine and cos. tan multiplied by cos will give us sine. So if three continuous functions are taken the product of the first and the third functions result in the function in between them. We can say that tan theta multiplied by cos theta equals sine theta. Take these three functions for example. This multiplied by this will give cosec. So sec theta multiplied by cot theta will give us cosec theta. Even cot multiplied by sine will give us cos. cosec multiplied by tan will give us sec and so on. First term multiplied by the third term will give us the term in between them. Does it end here? Not yet. Let's redraw the hexagon with one in the center. tan is equal to sine by cos, cot opposite to tan, cosec here as all c's on the right and what remains is sec. Now focus on the functions at diagonally opposite vertices. sine and cosec, cos and sec, cot and tan. They are reciprocals of each other. Look at this arrow for instance. This arrow tells us that sine theta equals one by cosec theta. What about this arrow? This arrow tells us that cos theta is one by sec theta. And this arrow tells us that cot theta equals 1 by tan theta. If the arrow is drawn like this, we can even say that cosec theta is reciprocal of sine theta. So remember. Functions at diagonally opposite vertices are reciprocals of each other. Be patient, it doesn't end here. But let's review what we have covered so far. This equals this by this. Or this equals this by this. This multiplied by this equals 1 or this multiplied by this equals 1. This multiplied by this equals this and this is equal to reciprocal of this. Let's draw the super hex again to know the other stuff it tell us. tan is equal to sine by cos, cot opposite to tan cosec here as all c's are on the right hand side and the remaining function sec here. Do you remember complementary angles? Yes two angles are complementary if their sum equals 90 degrees. And the trigonometric functions had a special relation with complementary angles. Look at this arrow. What does it tell us about sin and cos? It tells us that sin of theta equals. cos of 90 degrees minus theta. This horizontal arrow tells us that sin of an angle is equal to the cosine of its complementary angle. Similarly this arrow tells us that tan theta equals cot of 90 degrees minus theta. And this arrow tells us that sec theta equals cosec 90 degrees minus theta. And if the arrow is drawn this way it will tell us that cos of theta equals sin of 90 degrees minus theta. So a horizontal arrow tells us that one function of an angle is equal to the other function of its complementary angle. I think I have lost track of how many formulae the super hex has helped us remember and there's more. This is the super hexagon with a 1 in the center. tan is equal to sign by cos, cot opposite to tan, cosec is written here as we have all the C's on the right hand side, and the only remaining function sec is written here. Look at the figure. There are six triangles formed inside. Focus on three. This one, this one and this one. Let's zoom out slightly. Okay, so here we go clockwise within each of the triangles starting with top left position. Like this, sine, cos and one. This gives us one very important identity. This squared plus this squared equals this squared. So sine squared theta plus cos squared theta equals one. Now let's go clockwise in this triangle. This squared plus this squared will equal this square. One plus cot squared theta will equal cosec squared theta. If we go clockwise in this triangle we get tan squared theta plus one equals sec squared theta. Amazing, Isn't it? This arrow can also be drawn anti-clockwise from the bottom of the triangle like this. The only difference with anti-clockwise is that there will be a minus sign before the second term. So it will be this squared minus this squared equals this squared. So we can say that 1 minus cos squared theta will equal sine squared theta. Similarly if these two also became anti-clockwise, we get two more results. cosec squared theta minus cot squared theta equals 1. And sec squared theta minus 1 equals tan squared theta. Now you know why we call it super. One hexagon helped us with so many formulae in trigonometry. But hang on you have to practice this procedure well. Because there are a few things you need to remember here too. The position of the functions, the directions of arrows and so on. This video is not a substitute for all the other videos done before this. It's just an add on. Memorizing formulae can never be the solution. Make sure you understand all the formulae we derived in the previous videos.

What's a super hexagon? Does it really help us remember all the trigonometric formulae and identities we have seen so far? Yes it does. It's a simple hexagon which looks like this. All we need to do is join all the opposite vertices and that gives us three diagonals. We write the number one in the center. It's still difficult to believe that one hexagon can lead us to so many formulae. To know the formulae, we give the six function names to each of the six vertices. But there is a particular way in which we need to write it. We start writing from this vertex, go around the hexagon and end at this one. It's simple! We just draw a hexagon join its opposite vertices and write a 1 in the center. Now let's see, how we write the six functions at the six vertices. We just need to remember one formula which we have already seen. tan is equal to sine by cos. We have used this formula like a zillion times now. tan is equal to sine by cos. We start at this vertex and write tan, sine, cos in the clockwise direction. tan, sine, cos. Now three functions and three vertices
remain. We write cot opposite to tan. Remember co-tangent opposite to tangent. What remains is cosec and sec. If we draw a vertical line through the center of the hexagon, we will have three vertices on the left and three on the right. So to remember what's written on this vertex just remember that we have all functions starting with c on the right side. So we write a cosec here and what remains is sec. That's all you need to remember about how the super-hex has to be drawn. Three diagonals joining the opposite vertices, the number one in the centre
and the six function at the vertices. So let's draw the super-hex on a new page. A hexagon with three diagonals joining the opposite vertices. The number one at the centre, tan is equal to sign by cos, cot opposite to tan cosec here since all c's are on the right and what remains is sec. Are you ready? You're going to learn a whole set of trigonometric formulae. This video is fun but also very long as we will be covering most of the trigonometric formulae we have studied before. To understand the first set of formulae let's try going around the hexagon in the clockwise fashion. We go like this. This tells us that tan theta equals sine theta by cos theta. Well that's how we started making our hexagon. Now lets see the next three functions in clockwise direction. These three! This tells us that sine theta equals cos theta by cot theta The first one equals the second one by the third one. Take the next three functions clockwise. First equals second by third. Hence cos theta equals cot theta by cosec theta. And if we take the next three functions, we can say that cot theta equals cosec theta by sec theta. And we will get two more formulae if we go clockwise. Cosec is equal to sec by tan. And sec is equal to tan by sine. Wasn't that simple? One hexagon and six formulae without any effort. Great! But what if you forget that you had to go clockwise and you go anti-clockwise instead. Let's say you started like this, in the anti clockwise direction. Don't worry! The pattern still holds true. First will equal second by third. tan theta equals sec theta by cosec theta. First equals second by third. If we take these three functions anti-clockwise, we can write sine theta equals tan theta by sec theta. If these three, then cos theta equals sine theta by tan theta and the list goes on. Whether you go anti-clockwise or clockwise, first will always equal second by third. We can easily get three more here. One simple hexagon and twelve formulae. Is that all? No, there's more! Let's draw the super-hex on a new page to understand what else there is! We draw the super-hex and write a 1 in the centre. tan is equal to sine by cos, cot opposite to tan, as we have all c's on the right, we write a cosec here and what remains is sec. In the previous set of formulae we did not use the one which was written in the centre. Let's see why it is placed like this. If we multiply the functions at the opposite vertices, we get a 1. This multiplied by this equals 1. sine theta multiplied by cosec theta equals 1. This multiplied by this also gives us 1. cos theta multiplied by sec theta equals 1. And this multiplied by this also equals 1. tan theta multiplied by cot theta equals 1. So when functions are diagonally opposite vertices are multiplied, we get a 1. Wait, there's more. We draw the super-hex, right the number one at the center. tan equals sine by cos, cot opposite to tan and cosec here since all c's have to be on the right and what remains is sec. I repeat the positions again and again so it gets etched in your brain after watching the video. Take any three continuous functions in the hexagon like tan, sine and cos. tan multiplied by cos will give us sine. So if three continuous functions are taken the product of the first and the third functions result in the function in between them. We can say that tan theta multiplied by cos theta equals sine theta. Take these three functions for example. This multiplied by this will give cosec. So sec theta multiplied by cot theta will give us cosec theta. Even cot multiplied by sine will give us cos. cosec multiplied by tan will give us sec and so on. First term multiplied by the third term will give us the term in between them. Does it end here? Not yet. Let's redraw the hexagon with one in the center. tan is equal to sine by cos, cot opposite to tan, cosec here as all c's on the right and what remains is sec. Now focus on the functions at diagonally opposite vertices. sine and cosec, cos and sec, cot and tan. They are reciprocals of each other. Look at this arrow for instance. This arrow tells us that sine theta equals one by cosec theta. What about this arrow? This arrow tells us that cos theta is one by sec theta. And this arrow tells us that cot theta equals 1 by tan theta. If the arrow is drawn like this, we can even say that cosec theta is reciprocal of sine theta. So remember. Functions at diagonally opposite vertices are reciprocals of each other. Be patient, it doesn't end here. But let's review what we have covered so far. This equals this by this. Or this equals this by this. This multiplied by this equals 1 or this multiplied by this equals 1. This multiplied by this equals this and this is equal to reciprocal of this. Let's draw the super hex again to know the other stuff it tell us. tan is equal to sine by cos, cot opposite to tan cosec here as all c's are on the right hand side and the remaining function sec here. Do you remember complementary angles? Yes two angles are complementary if their sum equals 90 degrees. And the trigonometric functions had a special relation with complementary angles. Look at this arrow. What does it tell us about sin and cos? It tells us that sin of theta equals. cos of 90 degrees minus theta. This horizontal arrow tells us that sin of an angle is equal to the cosine of its complementary angle. Similarly this arrow tells us that tan theta equals cot of 90 degrees minus theta. And this arrow tells us that sec theta equals cosec 90 degrees minus theta. And if the arrow is drawn this way it will tell us that cos of theta equals sin of 90 degrees minus theta. So a horizontal arrow tells us that one function of an angle is equal to the other function of its complementary angle. I think I have lost track of how many formulae the super hex has helped us remember and there's more. This is the super hexagon with a 1 in the center. tan is equal to sign by cos, cot opposite to tan, cosec is written here as we have all the C's on the right hand side, and the only remaining function sec is written here. Look at the figure. There are six triangles formed inside. Focus on three. This one, this one and this one. Let's zoom out slightly. Okay, so here we go clockwise within each of the triangles starting with top left position. Like this, sine, cos and one. This gives us one very important identity. This squared plus this squared equals this squared. So sine squared theta plus cos squared theta equals one. Now let's go clockwise in this triangle. This squared plus this squared will equal this square. One plus cot squared theta will equal cosec squared theta. If we go clockwise in this triangle we get tan squared theta plus one equals sec squared theta. Amazing, Isn't it? This arrow can also be drawn anti-clockwise from the bottom of the triangle like this. The only difference with anti-clockwise is that there will be a minus sign before the second term. So it will be this squared minus this squared equals this squared. So we can say that 1 minus cos squared theta will equal sine squared theta. Similarly if these two also became anti-clockwise, we get two more results. cosec squared theta minus cot squared theta equals 1. And sec squared theta minus 1 equals tan squared theta. Now you know why we call it super. One hexagon helped us with so many formulae in trigonometry. But hang on you have to practice this procedure well. Because there are a few things you need to remember here too. The position of the functions, the directions of arrows and so on. This video is not a substitute for all the other videos done before this. It's just an add on. Memorizing formulae can never be the solution. Make sure you understand all the formulae we derived in the previous videos.

Transcript for:Memorizing Trigonometric Identities with Hexagon

Transcript for:
Memorizing Trigonometric Identities with Hexagon