Learn about Trig identities part 2 - parr 4 if you watch the proofs Video

By: psubob More than a year ago

intresting and must see for beginners

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Learn about Trig identities part 2 - parr 4 if you watch the proofs

Learn about Trig identities part 2—part 4 if you watch the proofs Now, I'm going to do a bit of a review of everything we’ve learned so far about maybe even Trigonometry and Trig Identities and then we will see if we can come up with the maybe use that we already know the couple of more trig identity. From SOH CAH TOA, we know that sine of Θ is equal to the opposite over the hypotenuse like if you were to draw a triangle here. So this is theta. This is the opposite, this is the adjacent, and this is the hypotenuse. Then sine of Θ is equal to opposite over hypotenuse plus sine of Θ. This is basic review hopefully this point is the adjacent over the hypotenuse. The tangent of theta is equal to the opposite over the adjacent which is also equal to the sine of Θ over the cosine of Θ. We showed this in a couple of videos ago and then these are kind of almost definitional but the cosecant of Θ is equal to the hypotenuse over the opposite which is the same thing as one over sine of Θ and you could just memorize this. I kind to find it silly that there is such a thing as cosecant. I guess it’s just for convenience because everyone knows it’s just one over sine of Θ and same thing for secant. If really for convenience instead of having to say you know in the case of secant. That’s one if you end up with equation one over cosine of theta becomes oh that’s just the secant of Θ and it can actually has some obvious properties if you are to draw unit circle and all that too so that’s equal to the hypotenuse over the adjacent which is equal to one over cosine of Θ and then of course cotangent of Θ is equal to the adjacent over the opposite which is equal to 1 over ten Θ. Of course, that’s also equal to cosine of Θ over sine Θ. It’s just the opposite of the tangent of theta right? Or that’s the same thing as what? That’s the sane thing as the secant, no, no, no it is the same thing as the cosecant of—I just want to get the inverse as well. Let’s prove it what is it actually. I was confusing myself so this is the same thing as one over the secant of Θ over one over the cosecant of Θ right? Secant of theta, cosecant of theta and then that equals the cosecant of theta over the secant of theta. I wouldn’t waste your time memorizing so we know that the cotangent if Θ is equal to one over tangent Θ is equal to the cosine over the sine and it also equals the cosecant over the secant. I would worry about really memorizing this. You could derive it if you have as you could tell I really didn’t have this memorize either. We also learned in previous videos that the sine squared Θ plus the cosine squared of Θ is equal to one that just comes from the Pythagorean Theorem and then if you play around with this a little bit, you’d also get that tangent squared Θ plus one is equal to the secant squared Θ. You actually go from here to here if you just divide both sides of this equation by cosine squared so we know that. Then if you’ve watched the last proof videos I made, we also know that the sine of let’s say a + b = sine of a times the cosine of b plus. Let me erase some of this because I don’t think that that is an important trig identity. You can derive it on your own. I just wanted to show you that you could figure it out using too much space. Now I have space and we find that blue color I was using and make sure that my pen is small. Okay so that the sine of a times the cosine of b plus the sine of b times the cosine of a. you might want to memorize this. This actually becomes really useful when you actually start doing your calculus because we have to solve the derivatives and tell the girls that you might have to now the identity and that’s not hard to memorize right? If the sine of one of them times the cosine of one of them plus the other way around, that’s all this is and then we also learn the cosine of a plus b is the cosine of both of them minus the sine of both of them. So that is equal to the cosine of a times the cosine of b and we’ve—I proved this another video hopefully to your satisfaction minus the sine a times sine of b. this is very useful because from this we can come up with the bunch of other trig identities. For example, what is the sine of 2a? Well, that’s just the same thing as the sine of a plus a. And if we used this trig identity up here, that is equal to sine of a cosine of a plus the sine of a cosine of a. I just used this sine of a plus b identity up here and I just pulled a and b are both a wand what is this equal? This is two terms that are just both sine of a cosine of a so that just equals two sine of a cosine of a so we now have derived another trigonometric identity that might be in the inside cover of your trig or actually your calculus book. Let me—and all of this actually I could draw square on all of them. Let’s do another one. Let’s figure out. It is actually once you have a bit of a library of trig identities, you can really just keep playing around and sing what else you can and I encourage you to do so. You’d be amazed on how many other trig identities that you could come up with. For example, let’s do cosine of 2a. Cosine of 2a is equal to cosine of a plus a. cosine of a plus a whet did we say? It is the cosine of both of the terms times each other minus the sine of both the terms so that equals cosine of a, cosine of a right? Cosine of a times cosine of a minus sine of a, sine of a. we solve and we decided and it was the cosine of a plus b identity minus sine of a so what is this? This is equal to cosine squared a minus sine squared a, that’s interesting and I could and then we could play around. This is interesting because this is the form a2 - b2 right? So that’s also the same thing as a + b times a - b so that’s the same thing as cosine of a plus sine of a times cosine of a minus sin of a. I don’t know this isn't really the trig identity. I'm just showing you could play with things. The cosine of 2a is equal to cosine of a plus sine of a times cosine of a minus the sine of a. the sum of the cosine and sine of a and then times the difference. That’s just interesting. I’m just showing you that what’s fun of my trigonometry is you can kindly keep playing around with it and actually that’s probably that is how all of the trig identities were discovered. Let’s say that we have you know. We want to figure out what cosine of negative a is. Let me draw a right triangle. It’s almost a right triangle and let’s say this angle is a. so negative a if a unit circle would look something like this. So cosine of a if we say that this side is the adjacent side, this is the hypotenuse, and this would still be the hypotenuse, and this is the opposite and this is the negative opposite. So cosine of minus a is equal to what? This is minus a so it’s the adjacent over the hypotenuse so it equals the adjacent over the hypotenuse which you just ay it’s h but that’s the same thing as cosine of a because cosine of a is also the adjacent over the hypotenuse. Alright I am almost out of time. Let me switch to a new video.

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Learn about Trig identities part 2—part 4 if you watch the proofs

Now, I'm going to do a bit of a review of everything we’ve learned so far about maybe even Trigonometry and Trig Identities and then we will see if we can come up with the maybe use that we already know the couple of more trig identity. From SOH CAH TOA, we know that sine of Θ is equal to the opposite over the hypotenuse like if you were to draw a triangle here. So this is theta. This is the opposite, this is the adjacent, and this is the hypotenuse. Then sine of Θ is equal to opposite over hypotenuse plus sine of Θ. This is basic review hopefully this point is the adjacent over the hypotenuse.

The tangent of theta is equal to the opposite over the adjacent which is also equal to the sine of Θ over the cosine of Θ. We showed this in a couple of videos ago and then these are kind of almost definitional but the cosecant of Θ is equal to the hypotenuse over the opposite which is the same thing as one over sine of Θ and you could just memorize this. I kind to find it silly that there is such a thing as cosecant. I guess it’s just for convenience because everyone knows it’s just one over sine of Θ and same thing for secant.

If really for convenience instead of having to say you know in the case of secant. That’s one if you end up with equation one over cosine of theta becomes oh that’s just the secant of Θ and it can actually has some obvious properties if you are to draw unit circle and all that too so that’s equal to the hypotenuse over the adjacent which is equal to one over cosine of Θ and then of course cotangent of Θ is equal to the adjacent over the opposite which is equal to 1 over ten Θ. Of course, that’s also equal to cosine of Θ over sine Θ. It’s just the opposite of the tangent of theta right? Or that’s the same thing as what?

That’s the sane thing as the secant, no, no, no it is the same thing as the cosecant of—I just want to get the inverse as well. Let’s prove it what is it actually. I was confusing myself so this is the same thing as one over the secant of Θ over one over the cosecant of Θ right? Secant of theta, cosecant of theta and then that equals the cosecant of theta over the secant of theta. I wouldn’t waste your time memorizing so we know that the cotangent if Θ is equal to one over tangent Θ is equal to the cosine over the sine and it also equals the cosecant over the secant.

I would worry about really memorizing this. You could derive it if you have as you could tell I really didn’t have this memorize either. We also learned in previous videos that the sine squared Θ plus the cosine squared of Θ is equal to one that just comes from the Pythagorean Theorem and then if you play around with this a little bit, you’d also get that tangent squared Θ plus one is equal to the secant squared Θ. You actually go from here to here if you just divide both sides of this equation by cosine squared so we know that.

Then if you’ve watched the last proof videos I made, we also know that the sine of let’s say a + b = sine of a times the cosine of b plus. Let me erase some of this because I don’t think that that is an important trig identity. You can derive it on your own. I just wanted to show you that you could figure it out using too much space. Now I have space and we find that blue color I was using and make sure that my pen is small.

Okay so that the sine of a times the cosine of b plus the sine of b times the cosine of a. you might want to memorize this. This actually becomes really useful when you actually start doing your calculus because we have to solve the derivatives and tell the girls that you might have to now the identity and that’s not hard to memorize right? If the sine of one of them times the cosine of one of them plus the other way around, that’s all this is and then we also learn the cosine of a plus b is the cosine of both of them minus the sine of both of them. So that is equal to the cosine of a times the cosine of b and we’ve—I proved this another video hopefully to your satisfaction minus the sine a times sine of b. this is very useful because from this we can come up with the bunch of other trig identities.

For example, what is the sine of 2a? Well, that’s just the same thing as the sine of a plus a. And if we used this trig identity up here, that is equal to sine of a cosine of a plus the sine of a cosine of a. I just used this sine of a plus b identity up here and I just pulled a and b are both a wand what is this equal? This is two terms that are just both sine of a cosine of a so that just equals two sine of a cosine of a so we now have derived another trigonometric identity that might be in the inside cover of your trig or actually your calculus book. Let me—and all of this actually I could draw square on all of them.

Let’s do another one. Let’s figure out. It is actually once you have a bit of a library of trig identities, you can really just keep playing around and sing what else you can and I encourage you to do so. You’d be amazed on how many other trig identities that you could come up with.

For example, let’s do cosine of 2a. Cosine of 2a is equal to cosine of a plus a. cosine of a plus a whet did we say? It is the cosine of both of the terms times each other minus the sine of both the terms so that equals cosine of a, cosine of a right? Cosine of a times cosine of a minus sine of a, sine of a. we solve and we decided and it was the cosine of a plus b identity minus sine of a so what is this? This is equal to cosine squared a minus sine squared a, that’s interesting and I could and then we could play around. This is interesting because this is the form a2 - b2 right? So that’s also the same thing as a + b times a - b so that’s the same thing as cosine of a plus sine of a times cosine of a minus sin of a.

I don’t know this isn't really the trig identity. I'm just showing you could play with things. The cosine of 2a is equal to cosine of a plus sine of a times cosine of a minus the sine of a. the sum of the cosine and sine of a and then times the difference. That’s just interesting. I’m just showing you that what’s fun of my trigonometry is you can kindly keep playing around with it and actually that’s probably that is how all of the trig identities were discovered. Let’s say that we have you know. We want to figure out what cosine of negative a is.

Let me draw a right triangle. It’s almost a right triangle and let’s say this angle is a. so negative a if a unit circle would look something like this. So cosine of a if we say that this side is the adjacent side, this is the hypotenuse, and this would still be the hypotenuse, and this is the opposite and this is the negative opposite.

So cosine of minus a is equal to what? This is minus a so it’s the adjacent over the hypotenuse so it equals the adjacent over the hypotenuse which you just ay it’s h but that’s the same thing as cosine of a because cosine of a is also the adjacent over the hypotenuse.

Alright I am almost out of time. Let me switch to a new video.

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