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Fundamental Trig Identities


Fundamental trig identities. So far we've talked about the three main trig functions, sine, cosine, and tangent. Those three are ratios but technically from the three sides of the SOHCAHTOA triangle, it's actually possibly to create six ratios. And so each of the six is a separate trig function, and really all six are important to know.

So we know three of them already. So let's take a look at the SOHCAHTOA triangle. There's our familiar SOHCAHTOA triangle, it happens to have an angle of 41 degrees, there's an opposite adjacent hypotenuse side. And so certainly, three of the ratios we can create are the familiar SOHCAHTOA ratios, but there are three more ratios we can create.

And here they are, cotangent is adjacent over opposite, secant is hypotenuse over adjacent, and cosecant is hypotenuse over opposite, and those are the six ratios all together. So wait a second, what are those names? Let's look at these names very carefully, here are the full names. So we've already talked about sine, cosine, and tangent.

And now we're talking about cotangent, secant, and cosecant. And notice the way they are listed here, if you can remember the three on the left, the three on the right, it's just the same name with co in front of it. So at least some of these names have their origin in geometric relationships. Let's talk about this for a minute. So now let's look at a circle, could be the unit circle.

Has a radius of 1, center at the origin. And so AB and CD are parallel to the y-axis. So we have two vertical segments there, AB and CD. And it looks like B is the point where that radius line intersects the circle, it continues on, and D looks like it's tangent to the circle where it crosses the x-axis.

Okay, so notice a few things, that in triangle OAB the triangle inside the circle OB, the radius is 1, and of course OA is the cosine and the AB is the sine, okay? So that's the familiar SOHCAHTOA ratio. Now look at triangle slightly bigger triangle OCD. And so this one is the one that starts at O, passes through B, all the way out to C, drops down to D, and comes back along the x-axis.

Well, in that triangle, OD is 1. And so that would mean that opposite CD over 1 equals the tangent, so the tangent equals CD. And it means that hypotenuse over adjacent OC over 1 is secant, so OC equals the secant. But here's the really cool thing about this diagram.

Notice that CD, the segment that has a length equal to the tangent is actually tangent to the circle. It passes the circle and touches it at one point, that is in fact a tangent line. Notice that OC, which is the secant, actually cuts through the circle, and so this is what's known in geometry as a secant line. And so that's why those two functions have those names, because one represents the length of a tangent segment and one represents the length of a secant segment.

And so if you're a very visual person, that might help you remember these things a little bit. Okay, sine and cosine are the most elementary trig functions, and we can actually express the other four in terms of them, and these are really important formulas to know. Tangent we can write as sign over cosine.

Cotangent we can write as cosine over sine, so notice those two are reciprocals, tangent and cotangent are reciprocals. Secant is the reciprocal of cosine, and cosecant is the reciprocal of sine. Notice that people get confused sometimes because they think the s and the s should go together, the c and the c should go together, they don't. Secant is the reciprocal of cosine.

Cosecant is the reciprocal of sine. So the test may give you one of those if you need it in a problem, but it may expect you to remember it as well, so it's really good to have those four memorized. Now on the first lesson on training, we mentioned the fundamental Pythagorean identity, cosine squared plus sine squared equals 1. Now that we have two more functions, we can also express the other Pythagorean identities.

One of them is tangent squared plus 1 equals secant squared. One of them is cotangent squared plus 1 equals cosecant squared. So the test quite likely would give you these equations if a problem required them, but they may serve as a shortcut or a way to confirm the answer. Another thing I'll say is, if you're planning to take calculus, I guarantee, I absolutely guarantee, that you need to know all three of these equations cold once you're in calculus.

So I'll say a few things about these. Of course you can blindly memorize them, but we don't recommend that, what we really recommend is understanding them. And so if you start with the one at the top, cosine squared plus sine squared equals 1, you could divide everything on both sides by cosine squared, you'd get the top Pythagorean identity at the bottom tangent and secant, or you could divide everything in cosine squared plus sine squared equals 1 by sine squared, and then you get the bottom one, cotangent squared and cosecant.

Ultimately, you could go back to the original SOHCAHTOA triangle with ABC and start with the Pythagorean theorem, A squared plus B squared equals C squared. You may remember that we got this top Pythagorean identity, cosine squared plus sine squared equals 1, we got that, from taking A squared plus B squared plus C squared and dividing everything all three terms by C squared. Let's say that dividing by C squared, we could divide all three terms by either A squared or B squared.

And if you do that and then sub in from the ratios, what the trig functions are, you'll produce these two Pythagorean identities. And so I strongly suggest do that on your own. Show in a couple different ways that you can come up with all these equations, because then you'll really understand them. Okay, now we can move onto a practice problem.

Pause the video and we'll talk about this. All right, in the triangle to the right, in terms of b and c, which of the following is the value of tangent theta? All right, well let's think about this. We have two sides there, we're given b and c. And of course c is the hypotenuse, b is the opposite, and tangent is opposite over adjacent.

We have the opposite, we don't have the adjacent, so we're gonna need that third side. Well, we can use the Pythagorean theorem. So the Pythagorean theorem tells us that b squared plus whatever the adjacent side squared is equal c squared, and we can solve this with the adjacent sign. Adjacent squared equals c squared minus b squared, take a square root of both sides.

Notice that taking a square root, we cannot take a square root of c and b separately, we have to leave it at that expression, c squared minus b squared. But that is an expression for the length of the adjacent sides, c squared minus b squared. Well now, now we're golden because tangent is opposite over adjacent. We have the opposite, we have the adjacent.

So, opposite over adjacent, and that would equal b over the square root of c squared minus b squared, and in fact that is answer c. We go back to the problem and we choose answer c. In summary, we introduce the other three trig functions, cotangent, secant, and cosecant. We discussed how to express the other four in terms of sine and cosine, so it's very good to understand how they fit into the SOHCAHTOA triangle, it's very good to understand how they're related to sine and cosine.

And finally, we discussed the three Pythagorean Identities.

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