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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 possible 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 this SOHCAHTOA triangle. There's our familiar SOHCAHTOA triangle, 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 opposite. And those are the six ratios altogether.

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're listed here. If you can remember the three on the left the three on the right is 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 AB is the sine. So that's the familiar SOHCAHTOA ratio. Now, look at triangle, slightly bigger triangle, OCD. And so this one is the one that comes, starts at O, passes through B all the way out to C.

Drops down to D and goes 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 sine over cosine. Cotangent we can write as cosine over sine. So notice that 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 cuz they think that 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 in the first lesson on trig, we mentioned the fundamental Pythagorean identity, cosine squared plus sine squared equals one.

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 will say is, if you are planning to take calculus I guarantee, I absolutely guarantee that you need to know all three of these equations cold once you are 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. Alternately, 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. Well, instead of dividing by c squared we can 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 on to 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 o 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 going to 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 equals c squared. And we could solve this with the adjacent side. 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 as that expression, c squared minus b squared. But that is an expression for the length of the adjacent side. c squared minus b squared. Well, 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 introduced 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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