WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For this lesson, we are going to continue on right triangles.
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We are going to go over ratios and right triangles.
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Trigonometric ratios: first of all, trigonometry is the study involving triangle measurement.
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Because we are going to go over trigonometric ratios, this all has to do with trigonometry.
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Now, I know that you are probably thinking that this is geometry, not trigonometry.
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But it involves a lot of trigonometry, because of the triangles.
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And we are going over right triangles, so for this section, and for the next couple of sections, we are going to be using a lot of trigonometric ratios.
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The most common trigonometric ratios are these three right here.
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The first one is sine; it is pronounced "sign"; this is how you spell it out: S-I-N-E...but we always write it as "sin," but it is pronounced "sign."
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The next one is cosine, but we shorten it as "cos"; but we call it "cosine."
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The next one is tangent, but we only write "tan."
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Now, one thing to remember about trigonometric functions: there are more than three, but these are the main ones that we are going to go over.
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These are the most common; and for now, we are only going to use these three.
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But these, we can only use with angle measures; it is very, very important to remember that we can only use
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the sine of an angle measure, the cosine of an angle measure, and the tangent of an angle measure.
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They never stand alone; they always have to go with an angle measure.
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So, if I say "the sine of 90 degrees," that is one way that I would say it; I am looking for the sine of 90 degrees.
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Or I can say "cosine of 45 degrees," "tangent of 60 degrees..."
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Always remember that sine, cosine, and tangent must be with an angle measure; you can only find the sine of an angle measure.
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These trigonometric functions don't stand alone; they are with angle measures, and only angle measures.
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You can't find the sine of just any random number; it has to be the sine of angle measure; that number must be an angle measure.
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So, if you have a scientific calculator, or if you have a calculator like this, you are going to need it.
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And we are going to practice finding values on the calculator.
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Here, trigonometric functions, again, are sine, cosine, and tangent.
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If you look on your calculator, you should have buttons that say "sin," "cos," and "tan."
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Now, these right here, inverse trigonometric functions: if you look at the same three buttons,
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then above it, it should say "sin^-1"; above the "cos," it says "cos^-1"; and above "tan," it says "tan^-1."
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Those are very, very important; we are going to practice using those key functions here.
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We have sine, cosine, and tangent; and then above it, the second key is inverse sine, cosine, and tangent.
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Now, when do we use each of these?--again, for these, we are going to use angle measures.
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So, if I wanted to find the sine of 60 degrees, then I would punch in "sin(60)"; and you are going to get a number.
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And that number says 0.866; so the sine of this angle measure equals this.
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That is when you use trigonometric functions: the sine, cosine, or tangent (depending on what you have to find)
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of the angle measure equals...and that is what your calculator is going to give you, the answer.
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Now, when do you use these? Well, if you punch in "sin" and any number right after sine--
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if you punch in sin, cos, or tan, and you punch in a number after sin, cos, or tan--
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the calculator is going to think--it is going to assume--that that number that you punched in is an angle measure,
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because again, you can only find those functions of angle measures.
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If I have, let's say, the sine of x equals 0.866--so I have the answer, and I am missing the angle measure--
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that is what I am missing, so that is where it has to go, here, because angle measure always has to go there--
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always, always, always, it is the sin of an angle measure--because I don't have the angle measure
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(I only have what it equals), I can't plug in this number here, because this is not the angle measure.
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So, if I punch in the sine of 0.866, the calculator is going to think that .866 is the angle measure, which it is not.
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This is the answer; I am looking for the angle measure.
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So, depending on what you have, you would have to use different things.
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Now again, if you punch in sin(60), the calculator will know that 60 is the angle measure,
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and therefore, the calculator is going to give you the answer to that, the sine of that angle measure.
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If you want the calculator to give you the angle measure (you are not giving the calculator the angle measure--
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you want the calculator to give you the angle measure), then that is when you have to use the inverse sine.
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You are telling the calculator, "Well, I have this--I have the answer 0.866;
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I don't have the angle measure to give you; given the answer, I want the angle measure."
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Then, you would, second, press the sine (on your screen, is should say that it is the inverse sine); and then you punch in 0.866.
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By doing that, the calculator now knows that that number that you punched in is not the angle measure.
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Then, the answer is 59.997, which is 60 degrees; so let's just write 60.
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Now, that is the angle measure; it is really important to remember which one you have to use.
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Whatever number you punch in after sine, cosine, or tangent needs to be the angle measure, and the calculator will assume that.
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So, if it is not the angle measure (you want the calculator to give you the angle measure), then you have to do inverse sine,
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so that the calculator will know that that number is not the angle measure (it needs to give you the angle measure).
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Let's just do a few of those...oh, before that, we are going to go over this right here, "Soh-cah-toa."
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Soh-cah-toa is just an easy way for you to remember three formulas.
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Here we have "soh"; "cah" is another formula, and so is "toa."
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Now, I know that it sounds funny; but just say it to yourself a few times, so that you get used to this word, "Soh-cah-toa."
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And that is going to help you remember three of the formulas, which are also known as the three ratios.
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Each of these stands for something: S is sine (we are going to write down this formula here).
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Sine of...again, it has to be an angle measure, so let's write...x, equals...o is for opposite, the side opposite;
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the opposite side, over...h is for hypotenuse; so all of this "soh" is this right here: "Sine of x equals opposite over the hypotenuse."
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That is the "Soh," and that is the ratio for sine.
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Now, for cosine, it is right here: "cah" is going to be "Cosine of x is equal to"...the a stands for adjacent side, over the hypotenuse for the h.
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Cosine of x equals the adjacent side over the hypotenuse side.
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And the last one, "toa" is for tangent: Tangent of x equals...o is for opposite side, over...a is for adjacent side.
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Then, these three right here are the actual trigonometric functions; and then the rest,
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the "oh," "ah," "oa," are all for sides: o is for opposite side; a is for adjacent; and h is for hypotenuse.
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So then, again, "soh" is "Sine of x equals opposite over hypotenuse."
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This one is "cosine of x equals adjacent over hypotenuse," and then, this one right here is "tangent of x equals opposite over adjacent."
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This will help you remember these three formulas; and that is what this was for.
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Sine of x equals opposite over hypotenuse; cosine of x equals adjacent over hypotenuse; and then, tangent of x equals opposite over adjacent.
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Again, x is going to be the angle measure--only an angle measure can go there.
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So, let's say C: we are going to find sine of C.
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That is C, so now we are talking about from this angle's point of view.
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From this angle's point of view, what is the side opposite?--because we are looking for the side opposite.
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The opposite side would be side AB; and "over hypotenuse"--what is the hypotenuse? BC, so it is over BC.
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So again, the sine of C (now, it doesn't always have to be the sine of C; they will either name the angle,
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or they will tell you from what angle's point of view), for this one, we are going to do angle C's point of view: what is the side opposite?
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It is that right there, over the hypotenuse, which is BC.
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And then, for the cosine of C, again, from angle C's point of view, the adjacent side is the side that is next to it--
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not that one that is opposite, but it is the one that is next to it; it is the other leg, the AC.
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The side adjacent to angle C is AC, over the hypotenuse, which is BC.
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And then, the tangent of C is going to be opposite (which is AB), over the adjacent, which is AC.
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So again, the sine of x equals opposite over hypotenuse; cosine of x equals adjacent over hypotenuse; tangent of x equals opposite over adjacent.
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This is really, really important to know; again, just say this a few times to yourself: "Soh-cah-toa."
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And that will definitely help you, because you do need to know these three ratios.
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Now, let's do a few practices on the calculator, finding trigonometric functions.
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We are going to find the value of each ratio, or the measure of each angle.
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The first one, the sine of 15 (now, this is an angle measure, so let me just do that--these are angle measures), equals...
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and that is what we are looking for: so you go on your calculator, and just punch in "sin(15)".
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And then, because you are writing "15" right after you punch "sin," the calculator knows that 15 is the angle measure.
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Then, the answer becomes .2588.
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And then, the next one: it will be the tangent of 72, and that becomes 3.0777.
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I am just rounding it to four decimal places: write that over here...3.0777.
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Now, here, we don't have the angle measure; the angle measure is e--that is what we want to find.
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We want the angle measure; so when we punch it in, we want the calculator to give us the angle measure.
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But if you punch in "tan(0.9201)," the calculator will think that this is the angle measure, which is not true.
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This is not the angle measure; e is the angle measure.
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So, you tell the calculator, "I am going to give you the answer, and I want you to give me the angle measure."
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Inverse tangent, remember, is 2nd and then tan: .9201...equals...
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and then, the calculator knows that you want the angle measure, and that is 42.6 degrees.
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The same thing here: we don't have the angle measure; we want the angle measure; that is what we are looking for.
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When you punch it in, you can't punch in cosine of this number, because then the calculator...
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now, let's just try it: just try clearing your screen, and then just punch in "cos(.2821)".
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Now, it is going to give you .999987 and so on; that rounds to 1.
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Now, this number is not the angle measure, because the calculator, because you punched in cosine of this number,
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would assume that this is the angle measure; and so, it is going to give you the answer if this were to be the angle measure.
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But what we have to do is tell the calculator that that number is not the angle measure.
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Convert it to inverse cosine, and then the calculator will give you the angle measure: 73.6 degrees.
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Just be very, very careful with that: if you have a number here, that would be the angle measure.
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If you have a variable there, then you are doing the opposite: you are looking for the angle measure.
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Make sure you punch in inverse trigonometric functions.
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The next one: Find sine of a, cosine of a, and tangent of a for each.
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Here is where we are going to be using Soh-cah-toa.
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This one right here is going to be "sin(a)," the angle measure, "equals"...don't forget: if you don't have the angle measure,
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make sure that you write a variable right there; you can't leave it as "sin() ="; there has to be something there.
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The sine of a equals opposite over hypotenuse; cosine of a is equal to adjacent over hypotenuse;
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and then, tangent of a is going to be opposite over adjacent.
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The first one: to find this, we are going to have to use the sine one: sine and sine.
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Then, sin(a) here is going to be opposite (from a's point of view, what is the side opposite?
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It is this, so what is the measure of that?)--it is 5, over the hypotenuse, which is 13.
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Now, you are just going to leave it like that, because it is a fraction, and you can leave fractions: sin(a) = 5/13.
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The next one, cosine of a: again, from A's point of view, it is going to be adjacent; there is the adjacent: 12,
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the one next to the angle, over the hypotenuse, which is 13.
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And then, tangent of a is opposite over adjacent: from this angle's point of view, opposite is 5; adjacent is 12.
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Now, the next one: let's look for a right here, again, from A's point of view.
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So then, sine of a is opposite (which is 3), over hypotenuse (which is 5).
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Cosine of a equals adjacent (which is 4), over the hypotenuse (which is 5).
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And then, tangent of a is opposite (3) over the adjacent (which is 4).
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That is all they wanted you to find--just those things.
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Now, if they wanted you to find the actual angle measure, that is different;
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then you would have to use your calculator and do the inverse sine,
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because again, this is not the angle measure, so you can't punch in the sine of this number.
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So, you are going to have to do inverse sine, so that you find the angle measure.
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But then, here, for this problem, you don't have to find the angle measure, because they just want you to find sin(a), cos(a), tan(a)--that is it.
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We found sin(a), cos(a), and tan(a) for each of these triangles; that is it.
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In the next example, it is going to ask you for the actual angle measure.
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So, you are going to have to use these trigonometric ratios to actually find missing sides and angles.
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You are actually solving for something there; in this, they just wanted you to actually just write down the ratio; that is it.
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Next, find the value of x.
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Like the previous example, we are not going to use...
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well, in the previous example, we used all three trigonometric ratios, because they wanted us to find sin(a), cos(a), and tan(a).
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For this one, we don't have to use all three; we just have to use whatever we need in order to find x.
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We have to first look for an angle's point of view.
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So, look for an angle: this is the angle that is given, so we are going to use this angle right here.
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And then, what sides of the triangle are we working with?
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Are we working with the opposite? No, we don't have the measure of the opposite one.
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We have the adjacent, and we have the hypotenuse.
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Now, if I were to write out Soh-cah-toa again, just so it is easier to see,
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the three ratios are going to be sin(x) = opposite/hypotenuse; the next one, cos(x), equals adjacent/hypotenuse;
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and then, the last one is tan(x), equal to opposite/adjacent.
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Which one of these three would we use?
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We don't have the opposite; we only have the adjacent, which is this side right here
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(because that is what we are looking for, so we have to include that one) and the hypotenuse.
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Which one uses, from this angle's point of view, the adjacent and the hypotenuse?
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Opposite over hypotenuse...no; adjacent over hypotenuse: so then, we are going to have to use this one right here; we are going to have to use cosine.
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Now, do we have the angle measure--do we have x?
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Yes, we do, so now we are going to just start plugging in these numbers for the angle measure, the side adjacent, and the hypotenuse.
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Cosine of 55 equals adjacent (what is the one adjacent? That is x), over (what is the hypotenuse?) 12.
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Now, we are going to go to our calculator, and we are going to find x.
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Cosine of 55: now, 55 is the angle measure, so I can just punch in cos(55); cosine of 55 is .5736.
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That equals...all of this is equal to this; that equals x/12.
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Now, how do I solve for x here?--I am going to have to multiply the 12: multiply 12 on that side, and then multiply 12 to this side.
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Then, x becomes...I just have to multiply: instead of clearing it, I can just leave that number, and then just multiply it to 12.
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And I get 6.8829: so, this right here, this length, is 6.8829.
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Again, to go over what we just did: cos(55)...now, why did we use cosine, and not sine and not tangent?
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It is because we have to look at what we have.
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And again, we don't have to use all three of them; we just have to use the one that we need,
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unless it is like the previous problem, where it asks for all three; this one is not--it is just asking to find the missing values.
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So, you have to look for cosine; you have to use cosine, because from this angle's point of view, you only have the adjacent and the hypotenuse.
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So, you are going to use cos(x) = adjacent/hypotenuse; cos(55) =...adjacent is x; hypotenuse is 12.
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You punch this into your calculator; you get this; then you multiply 12 to both sides, and you get the answer.
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Now, I can find this third angle measure by subtracting it by 180, or I can just do 90 minus this number, and then I get this number.
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I could do that, and then, if you want, you can use this angle's point of view.
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This right here is actually going to be 35 degrees; and then, instead of using 55, you can use this one.
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If you decide to use that one, then it is a different perspective, a different point of view.
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So then, what would you have to use?
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We have opposite (this is opposite), and then the hypotenuse; if you are going to use this angle, then you would have to use the sine,
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because sin(35) is opposite over the hypotenuse; so you have options.
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You don't have to do both; you just have to use one of them.
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Now, because this is the angle that is given, it would just be easier, instead of doing more work, to look for this angle, and then go on from there.
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But either way, that is an option for you, if you would like to just use that angle instead.
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Let's do the next one: here we have...
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And we are going to have to use this one; it is not like this one, where we have options,
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where we can use this angle or this angle; for this one, we have to use this angle, because that is the angle that we are looking for.
00:29:25.500 --> 00:29:31.200
And then, plus, we can't subtract it from 180 (because we don't know what this angle is) to find that angle.
00:29:31.200 --> 00:29:44.700
We have to use this angle; from this angle's point of view, you have to work with the side opposite and the hypotenuse.
00:29:44.700 --> 00:29:50.000
So, you have opposite, and you have hypotenuse; what are you going to use?
00:29:50.000 --> 00:30:01.400
With opposite and hypotenuse, you are going to have to use sine, because that uses the side opposite and the hypotenuse.
00:30:01.400 --> 00:30:05.700
The other ones use adjacent and the hypotenuse (no, we don't have the adjacent);
00:30:05.700 --> 00:30:13.700
this one uses opposite, which we have, but no adjacent; so we have to use the sine.
00:30:13.700 --> 00:30:19.700
Then, sine...and then, what would go as the angle measure?
00:30:19.700 --> 00:30:21.700
We don't have the angle measure; that is what we are looking for.
00:30:21.700 --> 00:30:34.900
So, we are going to put x right there; that equals...the side opposite is 21, over...what is the hypotenuse?...25.
00:30:34.900 --> 00:30:41.700
Then, from here, I am going to just go to my calculator.
00:30:41.700 --> 00:30:50.800
And again, don't punch in sine of this number, because then your calculator would think that this is the angle measure.
00:30:50.800 --> 00:31:05.500
But it is not; so you are going to do inverse sine; that is 2nd and sine; you should see that sin^-1.
00:31:05.500 --> 00:31:15.200
And then, 21 divided by 25...and then, you should get...
00:31:15.200 --> 00:31:21.400
That automatically just gives you x, because you already used the inverse sine.
00:31:21.400 --> 00:31:37.600
57.1 degrees: that is this angle measure; 57.1 degrees--that is x.
00:31:37.600 --> 00:31:45.600
That is it; just remember that you have to look for an angle.
00:31:45.600 --> 00:31:51.700
That way, you know from which angle's perspective you have to look to see what you have.
00:31:51.700 --> 00:31:58.200
And then, from there, based on what you do have, what sides are given, and what you are looking for,
00:31:58.200 --> 00:32:03.900
you are going to have to pick one of these: sine of x, cosine of x, or tangent of x.
00:32:03.900 --> 00:32:12.400
And then, using one of those ratios, you are just going to plug in the numbers, and then solve for whatever missing value you need to solve for.
00:32:12.400 --> 00:32:23.900
Let's do the last example: this one is a little bit tricky, but it is not difficult at all.
00:32:23.900 --> 00:32:33.000
Just remember that, when you are given points like this, and you have to find the measure of angle B,
00:32:33.000 --> 00:32:43.400
now, when you are given this problem, you don't know if you have to use trigonometric functions yet.
00:32:43.400 --> 00:32:51.800
So, let's just plug in the points first and see what we are dealing with.
00:32:51.800 --> 00:33:21.200
The first one, A, is (-1,-5), right there; and then, B is -6 (1, 2, 3, 4, 5, 6) and -5; and then, C is -1, 1, 2, right there.
00:33:21.200 --> 00:33:32.700
So, here is the triangle; I know that this is a right angle.
00:33:32.700 --> 00:33:48.200
I know that because here, this point was (-1,-5); this is -5; and this point right here was (-6,-5).
00:33:48.200 --> 00:33:59.000
So then, these two points are on the same y-value right there, so then it makes a horizontal line.
00:33:59.000 --> 00:34:10.300
And then here, since this is (-1,-5), and this is (-1,2), they share the same x-coordinate, so this is a vertical line.
00:34:10.300 --> 00:34:15.400
A horizontal and a vertical--they are perpendicular.
00:34:15.400 --> 00:34:27.200
Now, what was this point? This point was A; this was B; and then, this right here is C.
00:34:27.200 --> 00:34:36.400
And we want to find the measure of angle B; so then, this is the angle that we have to find.
00:34:36.400 --> 00:34:47.200
Now, this is our variable; and then, let's look for side length.
00:34:47.200 --> 00:34:57.300
Because here, it is perfectly horizontal or perfectly vertical, we wouldn't have to use the distance formula--we can just count.
00:34:57.300 --> 00:35:04.200
Now, when it comes to BC, I am going to have to use the distance formula if I want to find BC.
00:35:04.200 --> 00:35:13.100
Why?--because it is going diagonally, and you can't start counting to see the distance when it is going diagonally.
00:35:13.100 --> 00:35:20.300
When it is going horizontally or vertically, then you can; you can just count the units.
00:35:20.300 --> 00:35:36.500
So here, this is (-1,-5), and this one is (-6,-5); from -6 all the way to -5, this is 5 units.
00:35:36.500 --> 00:35:46.300
If you have a bigger graph, you can start counting: it would just be from here, 1, 2, 3, 4, 5; so this is 5.
00:35:46.300 --> 00:36:00.200
Then, this right here, the vertical--we can count that also, so here is -5: 1, 2, 3, 4, 5, 6, 7; this is 7.
00:36:00.200 --> 00:36:05.400
AB is 5; AC is 7; we don't know BC.
00:36:05.400 --> 00:36:10.500
Now, if you want, you can go ahead and solve for BC by using the distance formula.
00:36:10.500 --> 00:36:23.200
The distance formula is x₂, the second x, minus the first x, squared, plus (y₂ - y₁)².
00:36:23.200 --> 00:36:28.200
That is the distance formula; so you can just, using the coordinates for B and using the coordinates for C,
00:36:28.200 --> 00:36:34.400
just plug it into this formula to find the distance of B to C.
00:36:34.400 --> 00:36:49.500
But I don't think I will need it; now, let me re-draw this triangle...here is A, B, C.
00:36:49.500 --> 00:36:55.700
This is what I am looking for, here; this is 5, and this is 7.
00:36:55.700 --> 00:37:00.200
Now, if I am looking for this angle measure, what do I have to work with?
00:37:00.200 --> 00:37:11.400
I have the side opposite, which is 7, and I have the side adjacent.
00:37:11.400 --> 00:37:26.500
So, I know that I am going to use...Soh-cah-toa: from here, which one uses opposite and adjacent?
00:37:26.500 --> 00:37:30.400
That would be this one right here, so we are going to use tangent.
00:37:30.400 --> 00:37:39.800
That means that tangent of...I am just going to use b for the variable, because that is the angle that they want us to find...
00:37:39.800 --> 00:37:53.900
tangent of b equals opposite (what is the side opposite? It is 7), and...what is the side adjacent? 5.
00:37:53.900 --> 00:37:59.300
So then, here is my equation: tan(b) = 7/5.
00:37:59.300 --> 00:38:04.500
Then, you just go straight to your calculator.
00:38:04.500 --> 00:38:10.500
Now, remember: don't forget; don't push tan(7/5), because that is not the angle measure; b would be the angle measure.
00:38:10.500 --> 00:38:19.600
Push 2nd, tan, to get the inverse tangent, 7/5.
00:38:19.600 --> 00:38:33.100
I just close the parentheses; then, the calculator knows that I am giving them this answer, and that I want the angle measure.
00:38:33.100 --> 00:38:41.400
The angle measure that it gave me was 54.46 degrees.
00:38:41.400 --> 00:38:55.200
And that is it; now, when do we use Soh-cah-toa--when do we use trigonometric functions?
00:38:55.200 --> 00:39:04.500
First of all, you must have a right triangle; they must be right triangles.
00:39:04.500 --> 00:39:14.400
And #2: when you are dealing with angles and sides together (you are given an angle measure,
00:39:14.400 --> 00:39:19.700
but you are looking for a side, using that angle measure; or given sides, you have to look for an angle measure)--
00:39:19.700 --> 00:39:28.400
anything that uses a combination of angles and sides of a right triangle is when you are going to use Soh-cah-toa.
00:39:28.400 --> 00:39:37.000
What if they give you two sides, and they just want you to find the other side, the missing side, the third side?
00:39:37.000 --> 00:39:41.000
Well, we don't have to use Soh-cah-toa, because no angles are involved.
00:39:41.000 --> 00:39:57.500
It is just only the sides; so let's say they gave you that this is 5 and this is 7, and they wanted you to find the missing side.
00:39:57.500 --> 00:40:01.300
Now, do we have to use Soh-cah-toa here--do we have to use trigonometric functions here?
00:40:01.300 --> 00:40:08.700
No, because no angles are involved here; then what can we use?
00:40:08.700 --> 00:40:18.400
We can use the Pythagorean theorem, remember, because 5², a², plus b², equals c².
00:40:18.400 --> 00:40:25.800
Again, you are only using Soh-cah-toa when you are given sides, and then they want you to find the angle measure;
00:40:25.800 --> 00:40:31.200
or given the angle measure, to find the missing side; and so on.
00:40:31.200 --> 00:40:35.500
We are going to continue trigonometric functions next lesson.
00:40:35.500 --> 00:40:37.000
For now, thank you for watching Educator.com.