 So I recall from a previous video that we defined the six trigonometric ratios, sine, cosine, tangent, cosecant, secant, and cotangent of an angle theta with respect to any terminal point, any point on the terminal side, up to proportionality. It doesn't matter which point you choose as long as you don't choose the origin. No, no, no, you can't do that one. So consider this diagram here. We have a terminal point x comma y for some angle theta by the Pythagorean equation. Since every angle, if it's in standard position, does kind of form a right triangle, the distance between the origin and the point will call that r. It can be found using the Pythagorean equation x squared plus y squared equals r squared. There's no ambiguity going on right there. And so we defined the six trigonometric ratios using all the possible combinations of x, y, and r here. So to remind ourselves of some things we saw, notice we saw that sine theta was equal to y over r. We also saw that cosecant theta was defined to be r over y. You'll notice that these fractions right here are actually just the same fraction but flipped upside down, y over r versus r over y. These are known as reciprocals. If we take the reciprocal of sine, we actually get cosine, a cosecant, excuse me. And so sine and cosecant are reciprocals of each other. And this gives us the first of what we refer to as a trigonometric identity, which the first one right here, this is one of the three reciprocal identities. What we mean by identity is that it doesn't matter which angle you put in for theta, cosecant of theta will always equal one over sine theta for any choice of angle theta whatsoever because these ratios are defined to actually be reciprocals of each other. Another example of this, let's take cosine, cosine we defined to be x over r, secant was also defined to be r over x for which this establishes that these are also reciprocals of each other. Secant is just cosine upside down and likely cotangent and tangent are also reciprocals of each other. For secant, we originally defined to be y over x and then cotangent we defined to be x over y. So those are, excuse me, x over y. They're reciprocals of each other. So that's something you're going to want to remember. Sine and cosecant are reciprocals, secant and cosine are reciprocals, and tangent and cotangent are reciprocals with each other. All right, another one, so those are the so-called three reciprocal identities. The other trigonometric identities I want to introduce in this video, which are in the same vein, are the two so-called quotient identities. So if we were to take the quantity sine theta divided by cosine theta, notice this looks like y over r divided by x over r, which as you take a fraction divided by a fraction, this means multiply by the reciprocal, y over r times r over x, for which the r on top and the r on bottom cancel out, and we get the simpler ratio y over x. But this ratio is none other than just the tangent ratio. So we see that sine theta divided by cosine theta is equal to tangent theta, or more simply, we just say tangent is sine over cosine. A similar calculation can be done with cotangent. That is, if I were to take cosine theta over sine theta, this will look like x over r divided by y over r. Canceling this, we're going to end up with x over y, which is exactly the cotangent ratio. So we see that sine over cosine gives us tangent and cosine over sine gives us cotangent, which just makes sense right here, because if you know this quotient identity right here and you take its reciprocal, then you're just going to get cotangent, right? So you flip this fraction upside down, you get cotangent. And so these are some of the simplest and most elementary of the trigonometric identities. So notice from this here that if I know the sine ratio, which is negative three-fifths for a specific angle, and I know the cosine ratio, which is four-fifths, turns out I can compute all of the trigonometric ratios. Because by the quotient identity, which we see right here, we see that tangent theta is just going to equal negative three-fifths divided by four-fifths. Simplifying the fraction, this becomes negative three-fifths, excuse me, negative three-fourths there, because the one-fifth on top cancels with the one-fifth on the bottom. So we're going to get tangent of theta to be negative three-fifths. Cotangent of theta, we have actually two ways to compute that. Cotangent, we could just take cosine divided by sine, or we could just take the reciprocal of tangent and we end up with negative four-thirds. So I guess this tangent, that gives us cotangent. We can get secant of theta. Secant is the reciprocal of cosine, so we get five over four. And we can also get cosecant, which is the reciprocal of sine, so we get negative five-thirds. And so if we know sine and cosine, it turns out we know all six trigonometric ratios because of the quotient identity, sometimes called the ratio identities, or the reciprocal identities. So because of this perspective, it turns out sine and cosine are the most important of the six trigonometric ratios.