WEBVTT
Kind: captions
Language: en
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ok, so now we are going to learn some other stuff.
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So I guess we did law of sines last time?
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Now we will do the law of cosines.
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isn't that fun?.
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Ok, I can show you where this comes from.
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But you know I already showed you where the law of sines comes from.
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One derivation is enough
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Remember if you want to find the sides of a right triangle
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and you know an angle and side you can use SOH CAH TOA
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and thats what the trig ratios are for.
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If you have a non-right triangles you can use the law of sines.
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Law of sines if you remember.
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Basically you have a triangle that is not right.
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What we do is we label the three angle A, B, and C with capital letters.
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and we use the lower case a, b, c to stand for sides that are opposite to those angles
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Opposite, opposite, opposite.
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Ok? The law of sines says---
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For those of you missed class on Monday it was all review except for the law of sines part--
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The law of sines said that sine of any angle divided by its opposite side
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is equal to sine of the other angle divided by its opposite side and equal to the sine of the third angle divided by its opposite side.
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ok? Where are we able to use this?
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We are able to use the law of sines for certain triangles but not for other triangles.
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And where some information are been given
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For example, if you are given two of the sides and the included angle
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you would not be able to use the law of sines.
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So, when can I use the law of sines?
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You can use it if you have angle, side, angle (ASA) or you can use it if you have side, angle, angle (SAA).
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Do you know what I mean by that?
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Angle, side, angle. For example if I have angle A, side b, and angle C.
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One thing that we know is that the sum of interior angles of a triangle is 180 degrees.
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So if you know two of the angles you immediately know the third angle.
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In these two cases, given that you have two angles you can find the third angle.
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And since we have a pair angles and the sides you can find the other side.
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However, sometimes you don't have that information.
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Sometimes instead we have different sets of information.
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Then you need a second formula. ok?
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You will use this formula if you either have only sides, all three sides (SSS) and you have side, angle, side (SAS).
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Now, you use the law of cosines.
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I can derive the law of cosines, but you know.
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Not that exciting! the law of cosines says kind like the pythagorean theorem.
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It says c squared is equal to a squared plus b squared but since we don't have to have ,
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since we don't have a right triangle you have to make an adjustment.
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You have to subtract 2 a b cos(C).
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Thats the law of cosines. Let me take a picture of that!.
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I can put it up on America's favorite Instagram.
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That is a capital C that goes there.
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Let me make it easy to see!
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minus -- not plus! -- 2ab cos(C).
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That is our two laws.
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So you use these when you want to solve a triangle.
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In other words finding the missing information
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depending of what you have been given. So,
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Suppose I want to find c.
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The letters are arbitrary of course.
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You can call these any letters you want.
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If you want to find c, well we know that c squared is a^2 + b^2 - 2 ab cos(C).
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You get that from the law of cosines.
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So now you literarily just plug into the formula.
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c^2 is 10^2 -- Sorry 12 squared. Now 12^2 + 10^2 -2(10)(12)cos(60 degrees).
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This is more fun with the calculator of course.
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c squared is 144 + 100 -240 (1/2). Which is 244 - 120.
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Very important, these go together.
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ok? you don't do 244 -240. ok?
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People tend to do that mistake a lot.
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Remember this has to get multiplied by cosine first.
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ok? PEMDAS.
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This whole thing has to be figured out first.
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Alright, so c squared is 124. c is the square root of 124.
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Whatever the square root of 124 is.
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ok? Pretty straight forward right?
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Ok, say we've got it that way.
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see if you can figure out that angle C.
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Yes! [student] could it be a word problem?
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it could be word problem or it could be just a written problem like this where it would just say find the missing side.
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I will give you an example of word problem in just a minute.
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ok? So if you want to find angle C, what do we know?
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So, lets write the formula. c^2= a^2 +b^2 -2ab cos(C).
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So 11 squared is equal to 9 squared plus 8 squared minus 2 times 9 times 8 cosine C.
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We are trying to figure out what cosine of C is.
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121, 81, 64 minus 144 cos(C) .
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121 = 145 - 144 cos(C).
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Now as I said, don't do 145 -144.
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Thats is wrong!. ok?
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Remember 144 goes with the cosine of C.
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They multiply together.
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So now if you want to figure out what cosine of C is, we can subtract 145.
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We get -24 = -144 cos(C).
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Divide and you get 1/6
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So cosine C is 1/6.
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Cosine of what angle is 1/6? I have no idea! right?
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Do you know? Use your calculator.
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Or, you will say c is the inverse cosine of 1/6, and just leave it as this.
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ok? So if you were doing this on a test question, the answer will be the inverse cosine of 1/6.
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I will get back to that in a couple of minutes.
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We touched on the inverses back about a month ago, and now we'll spend a little more time on them.
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in about 5 minutes
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Do you understand what I did?
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It is just manipulating the formula.
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You make sure you are comfortable with that!
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Of course on web-assign you can use your calculator but you have to learn how to do without the calculator in the actual test.ok?
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So what if we had word problems.
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What kind of word problems would you get?
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Are we ready for this?
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Yes..? [student] How did I get -24?
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I took 145 and subtracted from both sides.
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I don't do like the teachers in high school where the put the number underneath and you add and subtract.
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It wasn't the way I was taught...
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so, I don't do that! Ok fine I am going to.
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I know you all were taught that way.
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It is a very bad way of learning this because then you became dependent on writing that way.
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but, whatever. It's not my job to fix 5th grade mathematics.
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What about word problems?
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What would a word problem look like?
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ok, so you go to a tower.
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You measure the distance and you say that tower is 1400 meters away.
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You use one of those cool laser things.
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If you look at a different tower, after having turned 30 degrees, and that tower is 1800 meters away.
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How far apart are the towers? This is actually how they measure how far away stars are. Ok. You measure the angle to where a star is
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then you move, and now you have a new angle.
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and measure angle to the star again.
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And then you can find out, since you know how far you mocws,
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you can find how far it is to where the star is-- that's called parallax, it is similar to this.
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Basically, you create a triangle.
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Ok, so lets think about what we did.
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You stood here.
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You said this tower is 1400 meters.
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this tower is 1800 meters this distance, this angle is 30 degrees.
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How far apart were those two towers?
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So take a minute and figure it out. Use the law of cosines.
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Alright, so if you want to figure out what x is, we use the law of cosines.
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The law of cosine says that x squared will equal to 1400^2 + 1800^2 -2 (1400)(1800)cos(30 degrees). ok?
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So x^2 is 1960000+ 3240000 - 5040000 times (square root of 3 over 2).
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Ok? Minus, minus. So x^2 is 5200000 (not that this really matters) - 2520000 radical 3.
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And whatever. ok? Did I get that right?
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Am I losing some zeros somewhere?
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Or did I gain a zero?
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One more zero, Here we go! ok? You can compute that, I don't care what the number is. Ok?
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Thats how you do with the law of cosines.
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Anyone wanna compute that for me?
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Make sure that you use degree, not radian mode on the calculator otherwise you will be embarrassed.
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could write a law of cosines program if it makes it easier.
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913.9 meters
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In SigFigs, 920 meters right? 910 meters.
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How many significant figures do I have? I've got 2.
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Yes, so really it is 910 meters.
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But this isn't a chemistry class.
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This isn't physics class.
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So you can through those out the window.
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It does not matter. Ok, none of that matters.
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Thats how you use word problems.
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Yes, [student] oh yeah, I mean this is good enough. ok?
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In fact, the truth is this is good enough with radical 3 over 2 substitute in.
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I don't really care if you square 1400.
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Yes, [student] oh I took 5040000 divided by 2 and multiplied with radical 3. You know that is one of those math things, ok?
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ok? Practice one more!
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Then we move on to inverse stuff.
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Alright that is long enough.
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For those who were paying attention when we did the law of coisnes formula.
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So one of the things we can do is looking at the information that you have and you try to use the law of sines or the law of cosines.
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Ok? How do we know when to use the law of sines?
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We could try to use the law cosines and doesn't work!
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We will be missing too many pieces of information.
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Ok? Or if you noticed if you have side, side, angle, angle .
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Side, angle, angle you use law of sines.
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So you will say (sin 60)/b is (sin 45)/10. ok?
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You cross multiply.
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you get 10 sin 60 = b sin 45.
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So 10 time radical 3/2 = b times radical 2 over 2.
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two cancels. (10radical 3) over radical 2 = b.
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You don't need to rationalize that.
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ok?
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Of course we don't give you angles if we don't know the sines and cosines
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You just need it in terms of sine and cosine. ok?
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because this is really 10 sin 60 over sin 45 equal b.
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So, if we gave me that angles, angles that you don't automatically know, you just leave it in terms of that.
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OK? Enough with this.
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Lets move on to some other stuff.
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ok, now we do inverse trig.
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We did a little of inverse trig last time now we will do a little more.
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So, we have trig right?
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when we have a thing function and an angle.
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For example I tell you that this is sin pi/6 = 1/2.
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Inverse trig is working backwards.
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So like you know the square root of 4 is 2 because you know that 2 squared is equal to 4.
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Ok? so, the sin pi/6 =1/2 is the same as saying pi/6, inverse sine of 1/2.
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Often written as arcsin 1/2. ok?
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So what is the inverse sine of 1/2 mean?
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The sine of what angle equals 1/2.
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That is what the inverse sine means in general.
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So if you know that sin x= A, x is the inverse sine of A.
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When we use that we don't divide, we reverse.
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We do the inverse of something.
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You cannot divide by sine.
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That sine is not multiplied by x.
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That is sin "of" x. ok?
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So you need the inverse function so kind of what we had before where we had the cosine of an angle equal to 1/6.
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So, we want to know what angle has a cosine of 1/6.
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And that is the inverse cosine of 1/6. ok?
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So that is what inverse trig is used for.
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So these is a complicated rule for inverse functions, inverse trig.
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Here is the problem.
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You know that sin pi/6 is 1/2 right?
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The sine of 30 degrees is 1/2. ok?
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So what is the inverse sine of 1/2?
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It si pi/6 because after all the sine of pi/6 is 1/2.
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But we could also say that it is 5pi/6.
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After all isn't sine of 5pi/6 equal 1/2?
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Yea! Sine 150 is 1/2.
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How about 13pi/6?
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Sure! The sine of 13pi/6 is 1/2.
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How about 17pi/6?
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In other words there is infinite number of angles whose sine is equal to 1/2.
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ok? So, we want to use what we call the principle angle.
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otherwise we get an infinite number of answers for something that it is not a function.
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The only one is pi/6.
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So how do we know what to use?
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OK? How do we know whats going on?
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Well, lets think about the problem.
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The problem is, think about the sine graph.
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The sine graph looks like that.
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So if you say where is equal to 1/2,
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we will have all these places. Just keeps going for ever.
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And, when we do the inverse perhaps we will fail the vertical line test.
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ok?Thats the problem that its going on.
00:24:07.080 --> 00:24:11.280
So we only want to use part of the sine graphs.
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Here and in here.
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Which is equivalent of between here and here on the original graph.
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So we only take a piece of the sine graph between here and here.
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When you do the inverse we will only get a single answer.
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Ok? so that is the theory of whats going on.
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In other words, when you want to do the inverse function
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we want to make sure that it will pass the vertical line test.
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So, we only use piece of sine graph so when you flip it,
00:24:47.160 --> 00:24:50.840
and you do the inverse will pass the vertical line test.
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We will restrict what happens with inverse sine, inverse cosine, all of those.
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Ok sin x, the domain is all real.
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The range is -1 to 1.
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Remember what the sine graph looks like.
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It only goes up to 1 and down to -1.
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If I want to do the inverse of a function I just switch the domain and range.
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So now the domain, x will be between -1 and 1.
00:25:51.720 --> 00:26:02.520
But I don't want the range to come to be all reals. so i restrict it and I say I am only going to take values -pi/2 to pi/2.
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In other words, only values in the first quadrant, quadrant I and quadrant IV.
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So if I ask you, you have a calculator, what is the inverse sine radical 3/2,
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you only get the answer of 60 degrees or pi/3.
00:26:24.320 --> 00:26:30.360
Ok? If I ask for inverse sine of radical 3/2.
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There is infinite number of angles that have sine radical 3 over 2.
00:26:35.080 --> 00:26:40.440
So I restricted only those angles that come out in the IV quadrant and I quadrant.
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That means that when I do the inverse, you get a function.
00:26:44.840 --> 00:26:48.280
So you do something similar for the inverse cosine.
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Inverse cosine, again I only do for values between -1 and 1.
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Cosine only goes up to 1 and down to -1.
00:26:59.600 --> 00:27:06.160
And now the restricted values are between 0 and pi.
00:27:06.880 --> 00:27:11.480
Why is the cosine between 0 and pi.
00:27:11.480 --> 00:27:23.480
Well, The cosine graph looks like that.
00:27:24.240 --> 00:27:31.040
The inverse cosine graph, looks like that!.
00:27:32.080 --> 00:27:39.440
This is 1 and this is -1. So we only take a piece between here and here.
00:27:39.440 --> 00:27:42.400
Because then it would not fail the vertical line test.
00:27:43.200 --> 00:27:46.800
It only goes from 0 to pi.
00:27:50.880 --> 00:27:56.600
That is why on the previous page I restricted the cosine between 0 and pi.
00:28:02.080 --> 00:28:07.640
alright! I am not going to do inverse tangent at the moment, but we will work with inverse sine and cosine for a couple of times.
00:28:07.880 --> 00:28:09.560
We will do some fun stuff!
00:28:12.920 --> 00:28:17.160
Again, remember what the inverse trig functions are all about.
00:28:22.360 --> 00:28:27.160
If you think about a trig function, plug in the angle you get out a number.
00:28:27.520 --> 00:28:31.520
So, if you plug the sine of an angle, you get out some decimal.
00:28:31.520 --> 00:28:34.640
So, if you plug the sine of a different angle, you get out a different decimal.
00:28:35.080 --> 00:28:39.800
So the inverse trig, if you plug in the decimal you get out an angle.
00:28:40.400 --> 00:28:43.600
So the inverse trig function gives you an angle.
00:28:44.200 --> 00:28:48.680
If I ask you for the sine inverse of x, what we are getting is an angle.
00:28:49.440 --> 00:28:59.120
ok? So what if I asked you sin( cosine-inverse of 2/5)?
00:29:00.400 --> 00:29:01.600
What does that mean?
00:29:02.320 --> 00:29:10.000
So, inverse cosine will be positive in the first quadrant.
00:29:10.480 --> 00:29:15.560
And remember the inverse cosine is an angle.
00:29:16.320 --> 00:29:22.400
And what it is? Is the angle that has a cosine of 2/5.
00:29:22.440 --> 00:29:23.560
So lets call it x.
00:29:23.560 --> 00:29:24.560
Because it does not matter.
00:29:25.480 --> 00:29:31.560
So inverse cosine of 2/5 means the cosine of some angle is 2/5.
00:29:32.600 --> 00:29:35.600
So, the inverse cosine of 2/5 is that x.
00:29:36.160 --> 00:29:38.320
Then, I just need to find the sine of x.
00:29:40.000 --> 00:29:44.200
How can I find the sine of x? SOH CAH TOA.
00:29:44.200 --> 00:29:45.440
Opposite over hypothenuse.
00:29:46.000 --> 00:29:49.280
So I can find the missing side with pythagorean theorem.
00:29:57.320 --> 00:30:02.320
So the sine of x is square root of 21 over 5.
00:30:03.440 --> 00:30:08.320
So, again this isn't hard, its just takes time to write down the concepts.
00:30:08.920 --> 00:30:12.040
If I ask you the sine, the inverse cosine of 2/5.
00:30:12.600 --> 00:30:20.080
What you do is you say, I am going to draw a triangle and put in an x and you know that the cosine of x is 2/5.
00:30:21.200 --> 00:30:23.200
Now, I am looking for the sin x.
00:30:24.560 --> 00:30:28.240
Well, once I know two sides, I can find the third side using pythagorean theorem.
00:30:28.480 --> 00:30:31.840
And now that I have three sides, I can find the sine.
00:30:32.760 --> 00:30:39.960
Ok? [students] What a great question!
00:30:49.720 --> 00:30:55.280
What if I said, what is the sine cosine inverse of -2/5?
00:30:56.920 --> 00:31:00.040
You say well, I am going to go in which quadrant?
00:31:00.040 --> 00:31:01.760
You go to the second quadrant.
00:31:01.760 --> 00:31:06.680
Because the inverse cosine and its either going to be in the first quadrant or the second quadrant.
00:31:06.680 --> 00:31:07.360
You go here.
00:31:09.320 --> 00:31:10.440
That is an angle.
00:31:11.000 --> 00:31:12.280
Use cosine of -2/5.
00:31:13.520 --> 00:31:17.600
If you do pythagorean theorem you get square root of 21 again.
00:31:18.560 --> 00:31:21.680
It has to be positive, because its pointing up.
00:31:22.720 --> 00:31:24.560
So I still want the sine of x.
00:31:24.800 --> 00:31:28.760
In this case it will be square root of 21 over 5.
00:31:28.760 --> 00:31:35.440
It would be exactly the same because, well that how the problem is. Ok?
00:31:41.880 --> 00:31:45.320
Lets do another one to make sure we get the concepts.
00:32:08.360 --> 00:32:14.480
Lets say I want to do the cosine sine inverse of 5/11.
00:32:15.440 --> 00:32:18.080
Well, that means that I have a triangle.
00:32:18.320 --> 00:32:22.000
And the sine inverse is positive so I am in the first quadrant.
00:32:25.240 --> 00:32:30.160
And the sine of that angle is 5/11.
00:32:33.040 --> 00:32:42.280
And now I just want to find cosine of x. ok?
00:32:42.880 --> 00:32:45.120
How do I find cosine of x?
00:32:45.120 --> 00:32:47.360
Well, cosine is adjacent over hypothenuse.
00:32:47.360 --> 00:32:49.120
I can just use pythagorean theorem here.
00:32:55.680 --> 00:32:57.920
or 4 radical 6.
00:33:05.960 --> 00:33:11.600
Yes, [student] Oh what did I choose that quadrant?
00:33:11.600 --> 00:33:14.880
because that makes the inverse sine of positive angle ok?
00:33:15.200 --> 00:33:20.880
So remember what I do to the inverse sine, up 90 degrees in the first quadrant,
00:33:20.880 --> 00:33:26.080
that would be positive answers and down 90 degrees to the fourth quadrant for negative answers.
00:33:26.080 --> 00:33:28.080
I am just giving you a good way to remember that.
00:33:34.760 --> 00:33:37.640
Now I am going to through in inverse tan now.
00:33:41.080 --> 00:33:43.480
Having inverse tan makes it easier.
00:33:52.840 --> 00:33:57.600
Ok, Inverse tangent of x is an angle whose tangent is equal to x.
00:34:10.360 --> 00:34:15.800
ok, The inverse sine, inverse cosine, inverse tangent.
00:34:16.640 --> 00:34:23.760
And doing over a positive value, I will always going to use first quadrant for the angle.
00:34:32.160 --> 00:34:38.720
In doing the inverse sine of a negative number, I will use the fourth quadrant.
00:34:38.720 --> 00:34:39.920
Except we have two choices.Right?
00:34:44.880 --> 00:34:46.480
Where is sine positive?
00:34:46.480 --> 00:34:49.200
The sine is positive in the first and second quadrant.
00:34:49.200 --> 00:34:50.960
And it is negative in the third and fourth quadrant.
00:34:50.960 --> 00:34:54.200
So if I want to do the inverse sine of an negative angle,
00:34:54.200 --> 00:34:57.880
I need to use the third quadrant angle or a fourth quadrant angle.
00:34:57.880 --> 00:35:01.080
So we never use the third quadrant.
00:35:01.080 --> 00:35:03.920
I am using the fourth quadrant angle for the inverse sine.
00:35:04.640 --> 00:35:07.520
For the inverse cosine, I have two choices;
00:35:07.520 --> 00:35:09.200
I can use the second or the third quadrant.
00:35:09.200 --> 00:35:10.960
Thats where the cosine is negative.
00:35:11.240 --> 00:35:12.680
I never use the third.
00:35:12.680 --> 00:35:14.920
So I am going to use the second quadrant.
00:35:16.400 --> 00:35:19.200
The tangent, I actually have two choices.
00:35:19.200 --> 00:35:21.920
And I am going to use the fourth quadrant.
00:35:22.280 --> 00:35:25.960
That has to do more on what the tangent graph looks like.
00:35:26.760 --> 00:35:29.200
So, Lets put that back up!.
00:35:34.840 --> 00:35:41.720
Ok, So in other ways I did inverse sine of a positive angle, a positive number.
00:35:41.880 --> 00:35:44.680
My angle is located in the first quadrant.
00:35:49.200 --> 00:35:58.040
If I had done the inverse sine of a negative five over eleven, I will get the angle down here.
00:36:03.080 --> 00:36:07.120
ok? That will still come to square root of 96.
00:36:08.960 --> 00:36:21.800
So this will still be (cos x= radical 96/ 11) ok?
00:36:23.200 --> 00:36:27.440
Lets practice a couple. They are not that bad!
00:36:29.280 --> 00:36:33.120
Alright! What about if I ask you for,
00:36:40.000 --> 00:36:46.280
what if I ask for the sine tan inverse of 2/7?
00:36:49.920 --> 00:36:57.120
Well 2/7 is a positive number, we go to the first quadrant and draw a triangle.
00:36:58.440 --> 00:37:02.760
And you know that the opposite side is 2 and the adjacent side is 7.
00:37:08.680 --> 00:37:15.960
then I can find the hypothenuse 2 squared plus 7 squared is 53.
00:37:17.120 --> 00:37:19.200
So that is the square root of 53.
00:37:24.440 --> 00:37:25.960
So, I want the sine of x.
00:37:29.200 --> 00:37:33.960
That is 2 over the square root of 53.
00:37:39.640 --> 00:37:50.200
You get it! [student] You want to know how I add these, I used pythagorean theorem.
00:37:50.200 --> 00:37:53.720
So, side squared plus side squared is the hypothenuse squared.
00:37:53.720 --> 00:37:58.040
[student] yes, because we had the hypothenuse we were looking for a side.
00:37:58.040 --> 00:38:01.240
ok? So this is a squared plus b squared equals to c squared.
00:38:07.240 --> 00:38:10.960
Why is in the first quadrant? Im a doing the inverse tan of a positive angle.
00:38:10.960 --> 00:38:14.000
I am doing the inverse trig function of positive number.
00:38:14.000 --> 00:38:16.120
So, if you are doing the inverse trig function,
00:38:16.120 --> 00:38:18.560
of any positive number I am always use the first quadrant angle.
00:38:19.400 --> 00:38:21.720
So lets practice a couple.
00:39:06.560 --> 00:39:09.360
Ok.there is three to play with!
00:39:09.800 --> 00:39:15.120
Alright! Lets do each of these.
00:39:15.120 --> 00:39:17.280
Then I will do little more explaining.
00:39:17.640 --> 00:39:25.280
Cosine sine inverse of -5/8. Negative 5/8.
00:39:25.280 --> 00:39:28.440
You say alright, Im doing the sine inverse of a negative angle.
00:39:29.080 --> 00:39:31.800
negative number, we say negative angle.
00:39:32.200 --> 00:39:34.520
So that means my angle is down here.
00:39:36.400 --> 00:39:40.520
The sine of that angle is -5/8.
00:39:44.320 --> 00:39:51.000
You use pythagorean theorem, 8 squared minus negative five squared is 39.
00:39:54.000 --> 00:39:55.440
The square root of 39.
00:39:56.240 --> 00:40:01.680
Therefore, this is the square root of 39 over 8.
00:40:10.560 --> 00:40:14.160
[students] this is 8 squared ok?
00:40:15.280 --> 00:40:25.920
[student] don't confuse when you are solving for the hypothenuse and when you are solving for a leg.
00:40:27.280 --> 00:40:33.440
ok? Second one.
00:40:36.200 --> 00:40:38.560
Tan cosine inverse of 3/7.
00:40:38.560 --> 00:40:39.920
So we are in the first quadrant.
00:40:39.920 --> 00:40:41.920
With a positive angle, positive number.
00:40:43.720 --> 00:40:46.120
So the cosine is 3/7.
00:40:48.920 --> 00:40:50.600
That is square root of 40.
00:40:53.080 --> 00:40:57.760
ok? And that is x.
00:40:58.440 --> 00:41:03.640
Tangent of x is square root of 40 over 3.
00:41:04.440 --> 00:41:08.280
We could ask you for the secant, cosecant, and cotangent.
00:41:08.840 --> 00:41:10.440
They all the same rules.
00:41:12.240 --> 00:41:19.680
Yes! [student] Sine inverse tan of -5/12. Negative 5/12 it will be down here in the forth quadrant.
00:41:23.560 --> 00:41:25.760
Tangent is -5/12.
00:41:25.960 --> 00:41:29.960
If you do pythagorean theorem it come out to 13.
00:41:32.760 --> 00:41:38.280
The sine is -5/13.
00:41:41.520 --> 00:41:44.640
You can think about a radical 169 if you wanted.
00:41:44.640 --> 00:41:50.040
If you have a low radical 60 or radical 82, or whatever you can lose a couple of points.
00:41:50.080 --> 00:41:53.280
So make sure that you do pythagorean theorem.
00:41:54.080 --> 00:41:56.040
You know by heart right?
00:41:56.040 --> 00:41:57.520
One of my favorite theorems!
00:41:59.520 --> 00:42:01.280
Pretty much the only one we know right!
00:42:01.680 --> 00:42:05.040
ok? why is inverse tangent in the fourth quadrant?
00:42:05.040 --> 00:42:09.240
Well, we really need to go over what the tangent graph looks like.
00:42:10.520 --> 00:42:13.160
You know that the sine graph looks like.
00:42:14.040 --> 00:42:16.760
You know what the cosine graph look like.
00:42:17.360 --> 00:42:22.920
Tangent graph kind ok looks like this.
00:42:30.640 --> 00:42:33.920
Then, more of these.
00:42:37.360 --> 00:42:40.880
And then repeats again.
00:42:53.600 --> 00:42:57.200
The tangent is periodic. ok?
00:42:57.200 --> 00:42:58.960
The tangent graph looks like that.
00:42:58.960 --> 00:43:01.840
So, we have the same problem we had for sine and cosine.
00:43:01.840 --> 00:43:03.640
So, if you pick some number.
00:43:04.680 --> 00:43:07.960
There is an infinite number of possible answers.
00:43:08.440 --> 00:43:11.960
So we have to restrict which answers do we want to use.
00:43:12.480 --> 00:43:16.640
Since we have the asymptote in the middle, of the x that we used,
00:43:17.000 --> 00:43:23.480
we are going to restrict just using the values of tangent that are closest to the origin.
00:43:24.480 --> 00:43:31.840
So when you graph inverse tangent functions,
00:43:45.280 --> 00:43:48.000
now we could, when we were flipping this,
00:43:48.000 --> 00:43:51.760
we could pile up the y -axis and down to the y -axis.
00:43:51.880 --> 00:43:57.360
So we only go from pi/2 to -pi/2.
00:43:57.360 --> 00:44:02.560
Thats why we only use values in the fourth quadrant and values in the first quadrant.
00:44:02.880 --> 00:44:04.840
ok? So Thats a range.
00:44:13.440 --> 00:44:16.920
domain is all reals.
00:44:17.600 --> 00:44:27.400
If you take the inverse tan of anything you want.
00:44:27.400 --> 00:44:28.320
Got the idea?
00:44:38.800 --> 00:44:39.760
So far so good?
00:44:41.680 --> 00:44:44.480
I've finished everything I need to do today.
00:44:44.480 --> 00:44:47.360
Do you mind if we end five minutes early?