Fundamental Trig Identities

Mike McGarry
Lesson by Mike McGarry
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Now we can talk about some of the fundamental trigonometric ratios. So far we've talked about the three main trig functions, sine, cosine and tangent. Those are certainly the most important three but technically from the three sides of the SOHCAHTOA triangle there's a total of six possible ratios. Each of the six is a separate trig function that you could need to know for the test.

We already know SOHCAHTOA, there they are the SOHCAHTOA ratios and the other three, cotangent which is adjacent over opposite, secant which is hypotenuse over adjacent and cosecant which is hypotenuse over opposite. The test could expect you to know these three as well. Here are the full names of all six just so you can see the spelling. Sine, cosine, tangent, cotangent, secant, and cosecant.

At least some of the names have their origin in geometric relationships. Think about this. Suppose we have a circle, so this would be part of a unit circle. Has a radius of one. AB and AC are parallel to the Y axis. So we're two similar right triangles, OAB and ODC.

Now in OAB ,OB is one, the hypotenuse. And this means that OA has a length that is the cosine of theta. And AB has a length that is the sine of theta. So that's an interesting little triangle in and of itself. We'll actually return to that triangle in a few videos from now. But the other triangles interesting for our purposes here.

Because in triangle OCD, OD, the horizontal leg, now has a length of one. And this means that CD has a length of tangent (theta). And OC, the hypotenuse, has a length of secant (theta). We'll notice that CD is a tangent line, it's a line tangent to the circle. And OC is a secant line, it's a line that cuts through the circle. That is the origin of those two names.

That's the origin of the name tangent and secant. Sine and cosine are the most elementary trig functions, and we can express the other four functions in terms of them. So tangent equals sine over cosine, cotangent equals cosine over sine, secant is the reciprocal of cosine, cosecant is the reciprocal of sine. The test may give one of these to you if you need it in a problem, but it may expect you to remember them.

It's good to know these well. Relatedly, every trig function can be expressed as the reciprocal of another. From the last slide we already saw that secant is the reciprocal of cosine. Cosecant is the reciprocal of sine. We can also add that tangent and cotangent are reciprocals of each other. Cosine is the reciprocal of secant and sine is the reciprocal of cosecant.

So sine and cosecant are reciprocals of each other, as our cosine and secant. You may have noticed that three of the six trig functions begin with the syllable co-. This is significant. Adding or subtracting the syllable co- changes the name of one of the trig functions into the name of what we call it's cofunction.

So sine and cosine are cofunctions of each other. So are tangent and cotangent. So are secant and cosecant. The meaning of this becomes clear in a right triangle. So here we have a right triangle and we have two angles, the red angle alpha, the blue angle beta.

And of course, they're complementary, they add up to 90 degrees. Now notice the following about the ratios. The ratio a over r, that would be the sine of alpha. The sine because it's the opposite over hypotenuse in the point of view of alpha. It would be the cosine of beta because it is the adjacent over the hypotenuse for beta.

Similarly, b over r would be the cosine of alpha and the sign of beta. The ratio a over b, would be opposite over adjacent for alpha, but adjacent over opposite for beta. And similarly, b over a would be the cotangent of alpha, or the tangent of beta. R over b would be the secant of alpha or the cosecant of beta.

And r over a would be the cosecant of alpha or the secant of beta. So notice each time whatever a function of an angle is, it equals the cofunction of the other angle in the triangle. The angle that is 90 minus the first angle. So we could say that any function of theta equals the cofunction of 90 minus theta. The view from the other angle in the triangle.

We've already seen that sine 30 equals cosine of 60 and sine 60 equals cosine 30. That's one example. This is part of the general cofunction pattern. Whatever sine of 17 degrees equals it must have the exact same value as cosine of 73 degrees, because 17 plus 73 equals 90. Those are two angles that add up to 90.

So any function of 17 degrees will equal the cofunction of 73 degrees and vice versa. The ACT will probably not directly test this idea, but it may be helpful. For example, if, in some multiple choice question, both sine of 17 degrees and cosine 17 degrees are listed as choices. Both have the same value so neither can be the correct answer since there is always only one correct answer so it would mean that you could eliminate both of them.

And even if you didn't know how to solve it would substantially raise your chances of guessing. In the first lesson on trigonometry, I mentioned the fundamental Pythagorean identity. And this is cosine squared plus sine squared equals one. That's a very important one.

Two similar equations, also known as Pythagorean identities are tangent squared plus 1 equals secant squared. And cotangent squared plus 1 = cosecant squared. Notice if you just remember the first of these two all you need to do is put co in front of the two functions to get the second of these two. So these two are cofunction versions of each other.

The ACT would quite likely give you these equations if a problem required them, but they may serve as a shortcut or a way to confirm an answer. Here's a practice problem. Pause the video and then we'll talk about this. Okay. So we're given a triangle and two sides are labeled, b and c.

And we want the tangent. And the trouble is, well tangent's opposite over adjacent. We have the opposite, but we don't have a letter for that adjacent side. So we're gonna have to express it in terms of b and c. Well possibility number one is to just think in terms of triangle. So this is a very visual geometric approach.

We know the, the opposite, we know the hypotenuse, we need the adjacent. We're going to use the Pythagorean theorem. Solve for the adjacent. So the adjacent equals that radical formulation. Then we're going to say tangent equals opposite over adjacent, so that's gonna be b over that radical, and so that gives us a value of tan theta.

So that's one way to get to an answer. And that would give us answer C. Now, there's a whole other way using Pythagorean identity. Some people will find this shorter. Some people will find this longer. It's important to appreciate both solutions, because when you can think about a problem two different ways, you'll really understand it.

So, now I'm gonna solve this same problem with Pythagorean identities. I'm gonna say well, one thing that I do know, I do know the cosecant. The cosecant is c over b. Well, I know that cotangent squared plus 1 equals the cosecant squared, so the cotangent squared must equal the cosecant squared minus 1, which is c squared over b squared minus 1.

Find a common denominator, c squared minus b squared over b squared. That's the cotangent squared, take a square root we get that, take a reciprocal and of course the reciprocal of the cotangent is the tangent, there we have an expression for the tangent. Again, some people might find that a longer way, some people might find it a shorter way.

It's good to have both solutions in your back pocket so you have different ways to attack a problem. Either way we can see that from among the five answers, C is the correct answer. Remember that it's always good to understand more than one way to solve a problem. You may find one method or the other faster or easier to understand.

One or the other might come naturally. And sometimes you can solve using one method and, if there's time, check your work with another method. And certainly as you're doing practice problems, even if you can solve it one way, then when your in an untimed situation reviewing the problem, its always good to check and make sure you could solve the problem the other way.

It gives you mental flexibility to have different ways to solve a problem. It's good to break the habit of thinking that there's only one way to solve any particular problem. Mathematical thinking always involves as many different perspectives and approaches as possible. It requires tremendous intellectual agility and flexibility.

In summary, we introduced the other three trig functions: cotangent, secant and cosecant. We discussed how to express the other four functions in terms of sine and cosine, and how to express each function as the reciprocal of another. We discussed how any trig function of theta would equal the cofunction of 90 minus theta, and we discussed all three Pythagorean Identities.

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