 So, welcome to the second session in the second module. In the first session we have talked about the importance of the sinusoid or the sine wave in the context of voltage generation. I might mention that the sine wave has an important place in other branches of engineering too. For example, the harmonic oscillator is closely connected to a sinusoid. In fact, I encourage you to look up more applications or contexts in which the sinusoid plays an important role in engineering and in physics. But what we intend to do in this session is to look at the mathematical properties of a sinusoid, which make it attractive from the point of view of signals and systems. The first interesting thing about sine waves is that when you add two sine waves of the same frequency, it gives you back another sine wave of the same frequency. Let us prove this formally. So, we have two sine waves. Let us call them x1t and x2t. Of course, we are talking about continuous time here. x1t is a1 say cos, now I am going to use capital omega, capital omega t plus phi 1. So, the angular frequency of the sine wave is the same in x1 and x2. So, here you have a2 cos phi 2 plus omega t. Now, let us add, in fact any linear combination, say alpha 1 times x1t plus alpha 2 times x2t which essentially becomes, now let me decompose x1 and x2. In fact, x1t can be written as a1 cos omega t times cos phi 1 plus a1 sine omega t times minus sine phi 1 or we could remove the plus sign here and put a minus sign. Similarly, x2t is the same thing but with a2 and phi 2. Now, let us combine terms. How do we combine terms? Essentially, we take the terms with cos omega t and the terms with sine omega t and separate them and aggregate with cos omega t a few terms and with sine omega t a few terms. Let us do that. So, that becomes, now I will take the terms with cos omega t first and then the terms with sine omega t. Now, we can give names to each of these terms and let us do that. So, let us call this whole thing, say p and let us call this whole thing q. So, this is p times cos omega t plus q times sine omega t and this is easily seen to be p squared plus q squared positive square root cos omega t plus another angle, let us call it beta and beta is essentially the negative of the tan inverse of q by p. Now, the algebra is alright. What is important here is that you can combine two sine waves of the same frequency and get back a sine wave of the same frequency. There is some kind of additivity concept. So, in fact, here we have gone a step further, we have shown this for a linear combination. So, there is some connection with linearity as you see. Sine waves have a very warm relationship with linear systems, as you can see and we are going to see more of that as we present. Anyway, there is another interesting thing about sine waves, when we differentiate a sine wave, we get back a sine wave of the same frequency, let us verify that. So, we have a sine wave, let us call it x t, A cos omega t plus phi and we take the derivative. It gives us minus A omega sine omega t plus phi. So, you see here we are, the derivative operation leaves us with a sine wave of the same frequency. A linear combination leaves us with a sine wave of the same frequency. And in fact, this is the reason why sine waves are such a favorite with electrical engineers. If you have a resistive, inductive and capacitive R, L, C circuit as you call it and if you have sine waves of a certain frequency appearing as sources in that circuit, all over that circuit, it is sine waves of the same frequency that are established. So, there is some kind of a self replicating behavior that sine waves either show or are compelled to show by virtue of these properties. Well, whatever it is, sine waves have two or rather three basic properties, the frequency but then we have agreed that the frequency is going to be held at a certain value when we talk about these two properties that we just mentioned, the amplitude and the phase, right. So, let us identify that. A sinusoid, so typical sinusoid looks like this, you know, the cos or the sine does not really matter. It does not matter whether I write cos omega t plus phi or sine omega t plus phi. It is essentially a matter of a different angle here. But what does matter is the other three attributes. What are the attributes of the sine wave? This is the amplitude. This is the phase and this is the angular frequency. Now, a change of amplitude in a sine wave is easy to understand. Let us see why. So, suppose you had the original sinusoid to be A cos omega t plus phi and the changed sinusoid to be B cos omega t plus phi and needless to say A and B are both positive real numbers. Then the changed sinusoid is equal to the original sinusoid multiplied by B by A, which is also a positive real number. So, it is very simple. The change, you see, the change of amplitude in a sinusoid is beautifully and simply described by just a multiplying constant. Now, why is this important? Because if the sinusoid describe the current in a resistor, for example, and the changed sinusoid denoted the voltage, essentially, you know, you could relate the current and the voltage by a constant called the resistance that constant is positive real. On the other hand, look at what happens when you change the phase. We will see in the next discussion that when we change the phase, we are going to have trouble. This property whereby we just have a multiplying factor describing the change is not true anymore and that would lead us to a very interesting alternate description of sinusoids leading us to what are called rotating complex numbers or phasors. So, see you again in the next discussion where we will talk about what happens when you change the phase and why then you need to go on to a different perspective for describing sinusoids. Thank you.