Identidades Trigonométricas | Identidades Recíprocas

How are you friends? I hope you are very well. Welcome to the identity course and now we are going to talk about reciprocal identities. In this video we are going to talk about two topics. First, how to learn reciprocal identities and in the second part of the video, in the second half, we are going to see, I am going to demonstrate why those identities exist. Well, first of all, how to learn reciprocal identities.

You must already know this order because you have already worked it a lot, I suppose you have already seen trigonometric reasons and have already talked about trigonometric functions. Well, then the angle, the name is the minus, it is the name that is placed, you can place sine of angle A or sine of theta or sine of A or sine of beta, I placed in this case the angle alpha.

So, you already know this order, right? First to sine, cosine, tangent, cotangent, secant and cosecant. In trigonometric reasons we already talked about these functions being inverses. We already knew that the sine is the inverse of the cosecant,

that the cosine is the inverse of the secant and that the tangent is the inverse of the cotangent. We only need to know this to know all the reciprocal identities.

First identity, the sine of alpha or the angle that you have set, for being the opposite of the cosecant, can be written as 1 over cosecant of the angle alpha. The angle must be the same, if you write A here, you have to write A or B and B or whatever, ready?

These are the inverses. The second, the cosine, so I write here that the cosine of alpha is equal, as the cosine is inverse of the secant, then we write here that it is equal to 1 over secant of alpha.

I'm not done with the video but I'm going to leave you an exercise so you can practice, you know you can pause the video, what you are going to do is now here you are going to find out or write down how much the tan alpha, cotangent alpha, secant alpha and cosecant alpha is.

So you are going to write these results and the answer will appear in 3, 2, 1. So if you did the exercise I guess you did well and with this you will be seeing that it is very easy to learn the identities. The tangent, because it is the opposite of the cotangent, it will be 1 over cotangent. The cotangent is the opposite of what? Of the tangent. Now we are going to return that, so cotangent is equal to 1 over...

tangent, the secant is the opposite of the cosine, so secant is 1/cosine and the cosecant is equal to 1/sen. We already know six identities, yes, of the reciprocal identities. Let's learn the other two. The other two are very simple. What are they?

the tangent is equal to these two above, that is, the tangent of alpha is equal to these two, sine over cosine.

if it is very easy to learn it and also because we no longer know the tangent the cotangent we can also know it cotangent of alpha as the tangent is or rather as the cotangent is the opposite of the tangent that we would write here the opposite then we no longer write sine over cosine but cosine over sine

and with this we can learn the eight reciprocal identities. We are going to move on to the second part, which is to demonstrate, for example, this: why tangent is equal to sine over cosine or all the reciprocal identities. So for that we have to know the following: So here we are going to remember what we saw in the trigonometric reasons.

You should already know that sine is equal to the opposite cathode over the hypotenuse, that cosine is the adjacent cathode over the hypotenuse, and that tangent is the opposite cathode over the adjacent cathode.

The cotangent being the opposite of the tangent, well, it is the opposite, right? And so with the secant and the cosecant. We already saw this in the, here I leave you the link of the course of trigonometric reasons. We are going to start demonstrating the eight identities that we no longer know. The first one I am going to demonstrate is this, that the sine of alpha is equal to 1 over cosecant of alpha. So, what am I going to do? I'm going to take this 1 over cosecant of alpha.

and I'm going to look to see if it is equal to sine of alpha, so what am I going to do? I'm going to change the word coccane by hypotenuse over the opposite cathode, so this would be 1 over and the coccane I can change it by hypotenuse over the opposite cathode.

Here to be able to make extremes and means this 1 I put a 1 in the denominator and multiply the extremes and the means. So here if I multiply extremes and means would be 1 by opposite cathode that is opposite cathode over 1 by hypotenuse that is hypotenuse. And what is the opposite cathode over hypotenuse? It is sine of alpha, that is, with this we check

that 1 over cosecant is equal to sine of alpha. We can do the same with all the other 5 if you want to do it. I'm going to check now the tangent. What was it? That tan alpha was equal to sine of alpha over cosine of angle, alpha or angle. How are we going to check it? I'm going to take this, sine of alpha over cosine of alpha.

and we will look to see if it is equal to the tangent, then the sine, I verify that it is, the sine is the opposite cathode over the hypotenuse, over the cosine, the cosine that is the adjacent cathode over the hypotenuse. We do exactly the same thing again, we multiply extremes and halves, and then here it will give me the extremes above, the opposite cathode by the hypotenuse, what is it? Well, it cannot be done, then the opposite cathode by

hypotenuse and below we write hypotenuse by adjacent cathode look here you can eliminate the hypotenuse with the hypotenuse and that we have left opposite cathode over adjacent cathode that is or that is opposite cathode over adjacent cathode that is tan alpha and here it is verified that sine over cosine is equal to

to tangent. As always, finally, I'm going to leave you an exercise for you to practice. You already know that you can pause the video. What you are going to do are two exercises. Well, in the next video we will see how to learn the Pythagorean identities, they are called like this, which are the ones that follow. Well, what you are going to do are two things. First, write what is equal to cosine of a, in this case the angle I wrote it as a, the tangent of a and the cosecant of a.

And in the second part, we are going to do it in the second part of the video, you are going to demonstrate that cotangent of B is equal to cosine of B over sine of B. And the answer will appear in 3, 2, 1. The truth is, for me to remember how this is done, I always have to write sine, cosine, tangent, cotangent, secant and cosecant. And with that you can do all this, right? If you can do it without doing that list, much better. But the truth is, I always do it.

So, cos = 1/sqrt , which must have placed the same angle. Tangent, you could have written either of the two. One that is equal to 1/cos , and the other that tangent is equal to sin /cos .

the other cosecant is equal to 1 over seno of A and here to demonstrate then we take this part which is the largest and we change coseno is adjacent cathode over hypotenuse, seno is opposite cathode over hypotenuse here we multiply extremes and halves adjacent cathode by hypotenuse above and below hypotenuse by opposite cathode the hypotenuse is eliminated and we have only adjacent cathode over opposite cathode that is

the cotangent. Well friends, I hope you liked the class. Remember that you can see the complete course of trigonometric identities available on my channel or in the link that is in the description of the video or on the card that I leave you here at the top. I invite you to subscribe, comment, share and like the video. And not being more, bye bye.