Once mathematicians developed trigonometric formulae for NON-right-angled triangles, it became apparent that we needed ratios for angles greater than 90° (because some triangles contain obtuse angles). This led to a redefining of the ratios.
If we superimpose a triangle (yellow) on a unit circle, you can see that its hypotenuse must be one unit long, since that is the radius of the circle. And, BECAUSE we have made the hypotenuse equal to one unit (and since the length of the hypotenuse is used in two of our main ratios … sine and cosine), an amazing thing happens!
This means that, if we draw a triangle IN THE UNIT CIRCLE, the length of the base (the adjacent side, the x-distance) is exactly equal in length to cos(θ) and the height of the triangle (the opposite side, the y-distance) is exactly equal in length to sin(θ)! In the video below I show how this works.
Now, ingeniously, we can do the same with the tangent ratio. Just draw a vertical line through x = 1 so that it is a tangent to the unit circle. Now enlarge the (yellow) triangle so that its base extends from the origin (0,0) to (1,0). Its altitude is now lying on the tangent to the circle. In this larger triangle it is the ADJACENT side (the base) that is now one unit long (and not the hypotenuse)! Therefore, in this larger triangle,
This means that, of we draw a triangle as described (with base of one unit), the length of the opposite side (in green) is exactly equal in length to tan(θ).
These become the new definitions of those three ratios … the lengths on such a graph rather than ratios in a right-angled triangle. In the video below I show how these new definitions allow us to calculate sine, cosine and tangent ratios for angles greater than 90°.
This redefining of the ratios led to a number of new discoveries and the world of igonometry opened up wonderfully and in surprising ways! You will be amazed to see where it leads.
What I have to share in the next four videos is a very graphic and logical way of illustrating, understanding and learning the four fundamental identities that we use in trigonometry.
What is contained in THIS video is related in every trigonometry textbook in one form or another. The contents of the two videos that follow, however, should be very new to you (although they do appear in some texts). Think of this video as establishing the foundation for the others ...
When we first learn trigonometry, we learn how the ratios relate to the sides of a right-angled triangle. By placing such a triangle within a circle (with a radius of one unit) on graph paper, however, new definitions emerge. I show what these new definitions are and how to understand them geometrically. With practice, you should be able to estimate the sine or cosine of any angle up to 360 degrees IN YOUR HEAD with 'reasonable' accuracy (within a decimal point or two). It is good to practise this skill. You can do so with a friend who can check your results on a calculator.
I then show how this new understanding gives rise to two of the four key identities ... tanθ ≡ sinθ/cosθ, and the first Pythagorean Identity, sin²θ + cos²θ ≡ 1.
At the conclusion of the video, I explain how you can quickly learn to reproduce the pattern/diagram and the two identities. It just requires that you invest a minute or two each day for a few days.
These new definitions mean that we have now broken free from the 'confines' of right-angled triangles and can now solve problems involving triangles of any shape! As you will learn, mathematicians discovered that they now had the tools to analyse wave motion and the motion of anything that oscillates or wobbles or moves in a repetitive way. What a revolution! I hope you enjoy these first steps along that journey.
It is something of a mystery to me that we learn three Pythagorean Identities in Trigonometry but do not learn what triangles they relate to! Doesn't that strike you as odd, too? It seems that almost all memory of these triangles has disappeared from our text books. These two videos (this, and the one following it) contain my attempts to correct that omission.
I believe that you will understand trigonometry much better if you learn this material. If you learn about each of the three triangles that I describe here, you will understand all six trigonomeric ratios in a new way. As a bonus, you will remember the three identies MUCH more easily.
In this video, I explain how to derive, understand and learn the identity tan²θ + 1 ≡ sec²θ. You will also learn an intuitive and graphical way of understanding and estimating the values of tanθ and secθ given any angle θ.
In my previous post I expressed amazement that we learn the trigonometric Pythagorean Identities and, in most cases, have no real concept of the triangles to which they refer. This third video represents part of my attempt to rectify the situation for you.
In this video, I explain how to derive, understand and learn the identity 1 + cot²θ ≡ cosec²θ. You will also learn an intuitive and graphical way of understanding and estimating the values of cotθ and cosecθ given any angle θ.
I know it may sound 'dry,' but I hope you find this material fascinating and adopt it ... so that the use of this particular triangle will enhance your understanding of trigonometry.
Why, oh why, do these videos not go viral :-)?
For those of you from the USA, here is the illustration showing the cosecant as "csc."
In my previous three videos I have described the four trigonometric identities in some detail.
In this summary video I explain how you can learn all four identities and develop a 'feel' for them in just a few minutes each afternoon for about a week. Of course, using them helps, too! I think the video speaks for itself so I will add no more here except to list the four identities for you:
tanθ ≡ sinθ/cosθ
sin²θ + cos²θ ≡ 1
tan²θ + 1 ≡ sec²θ
cot²θ + 1 ≡ cosec²θ
Don't forget to practise estimating the values of the six ratios given any angle θ.
I hope the ideas presented in this video help you master your trigonometry.
One of my YouTube subscribers requested this video. Thank you Dhanishan!
In this first video I show how to "read" some of the "hints" that can help you prove trigonometric identities. This is the identity that we are going to prove:
First of all, it is important to learn the fundamental four trigonometric identities:
tanθ ≡ sinθ/cosθ
sin²θ + cos²θ ≡ 1
tan²θ + 1 ≡ sec²θ
1 + cot²θ ≡cosec²θ (this last term is written as csc²θ in many countries)
The last three identities are known as the Pythagorean Identities. To see why, you might watch the previous videos in this set.
Once you know these four identities, numerous other relationships/identities can be derived from them. They have various uses ... from simplifying derivatives and integrals in calculus, to studying various sequences, investigating complex numbers and wave structures, evaluating π, simplifying many trigonometric calculations, and more.
When given an identity to prove ... such as this one ... look for clues such as the number of terms on each side of the identity sign. Also look for denominators and characteristic structures, such as the square of trignometric ratios (e.g. sin²θ). In this video I discuss some of the things to watch for that will help you decide HOW to go about proving the relationship.
One of my YouTube subscribers requested this video. Thank you Dhanishan!
In this second video I show how to "read" some of the "hints" that can help you prove trigonometric identities. This is the identity that we are going to prove:
You will see the importance of knowing the four fundamental identities really well. The main principle demonstrated here is that, if two terms have to be reduced to one, it usually means that we are adding or subtracting fractions and, therefore, matching denominators is a vital goal. By the way, the four identities are:
tanθ = sinθ/cosθ
sin²θ + cos²θ = 1
tan²θ + 1 = sec²θ
1 + cot²θ = cosec²θ (in some parts of the world, cosecθ is written cscθ)
The identity that I prove this time is: (1+sinθ)/cosθ + cosθ/(1+sinθ) = 2secθ
Thank you, Dhanishan, for providing me with this identity (even though you could already prove it). It is quite a clever little question.
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What I have to share in the next four videos is a very graphic and logical way of illustrating, understanding and learning the four fundamental identities that we use in trigonometry.
