4

ProveNdmn 464 Yn en 4 (k-1)...

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ProveNdmn 464 Yn en 4 (k-1)

Prove Ndmn 464 Yn en 4 (k-1)



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Prove Equation 4.

Yeah. And this question we're asked to prove the definition of the arco sign otherwise known as the inverse coastline. Let's make use of two other equations here that I've written down for you. Mhm. So let's start let's start by letting y equal see inverse hyperbolic coastline or the our cash. Therefore that means that X will be equal to the cost of why. And we can plug this in. Yeah, we can plug this into our definition of the pythagorean identity for the hyperbolic co sign. We'll have the X squared minus the cinch squared of why. I'll just use why here is equal to one. Therefore X squared plus one equals the cinch squared of why and therefore X squared plus one in the square root. There should be a minus. You should go through minus. Of course they should both be minus. Is that equals the hyperbolic sine. Uh huh. And of course we know that the cinch yeah. Uh huh. Is equal to e minus co sign. So we'll have that plus or minus the square root of X squared minus one will be equal to each. The east. The y minus the hyperbolic co sign of why? But we know that this hyperbolic coastline of Y is just X. So we can move it to the other side. So you have X plus or minus the square of X squared minus one. It's equal to eat the Y. And therefore by taking the log on both sides. And we can ignore the negative fruit because we're going to assume that we're in the domain. So we have that Y equals C. Inverse hyperbolic cosine of X is equal to the natural log of X plus the square root of x squared minus one. That's how you prove this question.

Here in this problem late biological, too, because at Amber's ex, so I can write X is equal to because actually, I let it be question one. Also, we all know that because 80 square X minus sign ach, square X is equal to one. So replacing X by y so I can write the value edge sign at square by is equal to because at the square by minus one, simplifying it further. I can write Sign h Y is equal to under route access Choir minus one. Let it be question number two. Now I can die sign at tax Plus because HX is equal to it to the products replacing X y Y sign h y plus cause h Y is equal to the to the power y. On further simplification, I can like the value is under good access square minus one plus X is equal to eat to the bar y so I can write it to the power Y is equal to X plus under root access square minus one. So I get the value of Y is equal to learn X plus under dude access square minus phone, simplifying it for the so I get to evaluate because action was X physical. Too Long X plus under good access squared minus one. So this is our answer.

All right this time were given an equal, angular triangle. And we just want to prove that that's also equal Adira prison since two. Too bad six. A little bit of explain. Okay, so he started are given. It's a angled The IHS grow to angle. He is congruent to angle f Let's give him And I just picked the triangle from, ah, the actual core alert that this came from You can pick whatever letters you want, not a big deal. Okay, so first step since the 1st 2 given our d e and E f let's start there. So how can we show d e? Yeah, working group? Well, since the angles opposite to them are congruent, we can use the converse of the sausages triangle theory to show these two sides in being So we can say me is congruent to e f. This is the converse the sauce Elise triangle There it's number steps just in case. So step two to step one, okay, And then we can do that again for e f and F d special. You know, it's people like flu e f and F d. So these two have to be the same for the exact same reason. Converse I sauces trying their step three. We can say he f isn't if promise is congruent to d f for the same reason Converse So sleaze triangle there. Okay, so we have the ian yet we have e f and D EFS or commonality. Here is e f, which means I can replace this e f with d e so we can states to purple. But just substituting around instead of e f. I can call that e His king grew into D f and that's just a little bit of simple substitution and that waas are thirds parasites here d e and D f So now we can say since we have all three pairs of the same, we can finish this off Vice going toe are trying to prove which is t e is congruent Teoh Uh, e f is congressional Teoh f d r more simply or triangle is equal lateral and this is just the transitive property of immigrants. And there you go. There is one more proof we'll finish


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