Geometry Ratios in Right Triangles

Welcome back to Educator.com.0000

For this lesson, we are going to continue on right triangles.0003

We are going to go over ratios and right triangles.0006

Trigonometric ratios: first of all, trigonometry is the study involving triangle measurement.0010

Because we are going to go over trigonometric ratios, this all has to do with trigonometry.0024

Now, I know that you are probably thinking that this is geometry, not trigonometry.0030

But it involves a lot of trigonometry, because of the triangles.0034

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.0041

The most common trigonometric ratios are these three right here.0051

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."0056

The next one is cosine, but we shorten it as "cos"; but we call it "cosine."0071

The next one is tangent, but we only write "tan."0082

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.0089

These are the most common; and for now, we are only going to use these three.0100

But these, we can only use with angle measures; it is very, very important to remember that we can only use0106

the sine of an angle measure, the cosine of an angle measure, and the tangent of an angle measure.0114

They never stand alone; they always have to go with an angle measure.0121

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.0124

Or I can say "cosine of 45 degrees," "tangent of 60 degrees..."0138

Always remember that sine, cosine, and tangent must be with an angle measure; you can only find the sine of an angle measure.0149

These trigonometric functions don't stand alone; they are with angle measures, and only angle measures.0159

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.0167

So, if you have a scientific calculator, or if you have a calculator like this, you are going to need it.0175

And we are going to practice finding values on the calculator.0186

Here, trigonometric functions, again, are sine, cosine, and tangent.0194

If you look on your calculator, you should have buttons that say "sin," "cos," and "tan."0200

Now, these right here, inverse trigonometric functions: if you look at the same three buttons,0209

then above it, it should say "sin^-1"; above the "cos," it says "cos^-1"; and above "tan," it says "tan^-1."0217

Those are very, very important; we are going to practice using those key functions here.0230

We have sine, cosine, and tangent; and then above it, the second key is inverse sine, cosine, and tangent.0240

Now, when do we use each of these?--again, for these, we are going to use angle measures.0250

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.0256

And that number says 0.866; so the sine of this angle measure equals this.0276

That is when you use trigonometric functions: the sine, cosine, or tangent (depending on what you have to find)0291

of the angle measure equals...and that is what your calculator is going to give you, the answer.0298

Now, when do you use these? Well, if you punch in "sin" and any number right after sine--0302

if you punch in sin, cos, or tan, and you punch in a number after sin, cos, or tan--0311

the calculator is going to think--it is going to assume--that that number that you punched in is an angle measure,0318

because again, you can only find those functions of angle measures.0324

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--0333

that is what I am missing, so that is where it has to go, here, because angle measure always has to go there--0345

always, always, always, it is the sin of an angle measure--because I don't have the angle measure0351

(I only have what it equals), I can't plug in this number here, because this is not the angle measure.0357

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.0365

This is the answer; I am looking for the angle measure.0378

So, depending on what you have, you would have to use different things.0382

Now again, if you punch in sin(60), the calculator will know that 60 is the angle measure,0385

and therefore, the calculator is going to give you the answer to that, the sine of that angle measure.0393

If you want the calculator to give you the angle measure (you are not giving the calculator the angle measure--0400

you want the calculator to give you the angle measure), then that is when you have to use the inverse sine.0407

You are telling the calculator, "Well, I have this--I have the answer 0.866;0414

I don't have the angle measure to give you; given the answer, I want the angle measure."0421

Then, you would, second, press the sine (on your screen, is should say that it is the inverse sine); and then you punch in 0.866.0427

By doing that, the calculator now knows that that number that you punched in is not the angle measure.0442

Then, the answer is 59.997, which is 60 degrees; so let's just write 60.0448

Now, that is the angle measure; it is really important to remember which one you have to use.0468

Whatever number you punch in after sine, cosine, or tangent needs to be the angle measure, and the calculator will assume that.0474

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,0482

so that the calculator will know that that number is not the angle measure (it needs to give you the angle measure).0488

Let's just do a few of those...oh, before that, we are going to go over this right here, "Soh-cah-toa."0495

Soh-cah-toa is just an easy way for you to remember three formulas.0507

Here we have "soh"; "cah" is another formula, and so is "toa."0518

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."0530

And that is going to help you remember three of the formulas, which are also known as the three ratios.0537

Each of these stands for something: S is sine (we are going to write down this formula here).0549

Sine of...again, it has to be an angle measure, so let's write...x, equals...o is for opposite, the side opposite;0558

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."0570

That is the "Soh," and that is the ratio for sine.0592

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.0600

Cosine of x equals the adjacent side over the hypotenuse side.0627

And the last one, "toa" is for tangent: Tangent of x equals...o is for opposite side, over...a is for adjacent side.0634

Then, these three right here are the actual trigonometric functions; and then the rest,0659

the "oh," "ah," "oa," are all for sides: o is for opposite side; a is for adjacent; and h is for hypotenuse.0666

So then, again, "soh" is "Sine of x equals opposite over hypotenuse."0678

This one is "cosine of x equals adjacent over hypotenuse," and then, this one right here is "tangent of x equals opposite over adjacent."0687

This will help you remember these three formulas; and that is what this was for.0697

Sine of x equals opposite over hypotenuse; cosine of x equals adjacent over hypotenuse; and then, tangent of x equals opposite over adjacent.0703

Again, x is going to be the angle measure--only an angle measure can go there.0731

So, let's say C: we are going to find sine of C.0735

That is C, so now we are talking about from this angle's point of view.0748

From this angle's point of view, what is the side opposite?--because we are looking for the side opposite.0752

The opposite side would be side AB; and "over hypotenuse"--what is the hypotenuse? BC, so it is over BC.0762

So again, the sine of C (now, it doesn't always have to be the sine of C; they will either name the angle,0777

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?0785

It is that right there, over the hypotenuse, which is BC.0796

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--0802

not that one that is opposite, but it is the one that is next to it; it is the other leg, the AC.0813

The side adjacent to angle C is AC, over the hypotenuse, which is BC.0821

And then, the tangent of C is going to be opposite (which is AB), over the adjacent, which is AC.0830

So again, the sine of x equals opposite over hypotenuse; cosine of x equals adjacent over hypotenuse; tangent of x equals opposite over adjacent.0845

This is really, really important to know; again, just say this a few times to yourself: "Soh-cah-toa."0860

And that will definitely help you, because you do need to know these three ratios.0867

Now, let's do a few practices on the calculator, finding trigonometric functions.0879

We are going to find the value of each ratio, or the measure of each angle.0888

The first one, the sine of 15 (now, this is an angle measure, so let me just do that--these are angle measures), equals...0893

and that is what we are looking for: so you go on your calculator, and just punch in "sin(15)".0905

And then, because you are writing "15" right after you punch "sin," the calculator knows that 15 is the angle measure.0916

Then, the answer becomes .2588.0923

And then, the next one: it will be the tangent of 72, and that becomes 3.0777.0936

I am just rounding it to four decimal places: write that over here...3.0777.0953

Now, here, we don't have the angle measure; the angle measure is e--that is what we want to find.0965

We want the angle measure; so when we punch it in, we want the calculator to give us the angle measure.0974

But if you punch in "tan(0.9201)," the calculator will think that this is the angle measure, which is not true.0979

This is not the angle measure; e is the angle measure.0987

So, you tell the calculator, "I am going to give you the answer, and I want you to give me the angle measure."0990

Inverse tangent, remember, is 2^nd and then tan: .9201...equals...1001

and then, the calculator knows that you want the angle measure, and that is 42.6 degrees.1011

The same thing here: we don't have the angle measure; we want the angle measure; that is what we are looking for.1027

When you punch it in, you can't punch in cosine of this number, because then the calculator...1033

now, let's just try it: just try clearing your screen, and then just punch in "cos(.2821)".1039

Now, it is going to give you .999987 and so on; that rounds to 1.1057

Now, this number is not the angle measure, because the calculator, because you punched in cosine of this number,1067

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.1078

But what we have to do is tell the calculator that that number is not the angle measure.1088

Convert it to inverse cosine, and then the calculator will give you the angle measure: 73.6 degrees.1101

Just be very, very careful with that: if you have a number here, that would be the angle measure.1111

If you have a variable there, then you are doing the opposite: you are looking for the angle measure.1118

Make sure you punch in inverse trigonometric functions.1126

The next one: Find sine of a, cosine of a, and tangent of a for each.1133

Here is where we are going to be using Soh-cah-toa.1140

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,1153

make sure that you write a variable right there; you can't leave it as "sin() ="; there has to be something there.1163

The sine of a equals opposite over hypotenuse; cosine of a is equal to adjacent over hypotenuse;1169

and then, tangent of a is going to be opposite over adjacent.1184

The first one: to find this, we are going to have to use the sine one: sine and sine.1195

Then, sin(a) here is going to be opposite (from a's point of view, what is the side opposite?1201

It is this, so what is the measure of that?)--it is 5, over the hypotenuse, which is 13.1218

Now, you are just going to leave it like that, because it is a fraction, and you can leave fractions: sin(a) = 5/13.1228

The next one, cosine of a: again, from A's point of view, it is going to be adjacent; there is the adjacent: 12,1235

the one next to the angle, over the hypotenuse, which is 13.1251

And then, tangent of a is opposite over adjacent: from this angle's point of view, opposite is 5; adjacent is 12.1257

Now, the next one: let's look for a right here, again, from A's point of view.1276

So then, sine of a is opposite (which is 3), over hypotenuse (which is 5).1281

Cosine of a equals adjacent (which is 4), over the hypotenuse (which is 5).1296

And then, tangent of a is opposite (3) over the adjacent (which is 4).1306

That is all they wanted you to find--just those things.1318

Now, if they wanted you to find the actual angle measure, that is different;1320

then you would have to use your calculator and do the inverse sine,1326

because again, this is not the angle measure, so you can't punch in the sine of this number.1330

So, you are going to have to do inverse sine, so that you find the angle measure.1335

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.1340

We found sin(a), cos(a), and tan(a) for each of these triangles; that is it.1347

In the next example, it is going to ask you for the actual angle measure.1354

So, you are going to have to use these trigonometric ratios to actually find missing sides and angles.1359

You are actually solving for something there; in this, they just wanted you to actually just write down the ratio; that is it.1366

Next, find the value of x.1375

Like the previous example, we are not going to use...1383

well, in the previous example, we used all three trigonometric ratios, because they wanted us to find sin(a), cos(a), and tan(a).1387

For this one, we don't have to use all three; we just have to use whatever we need in order to find x.1395

We have to first look for an angle's point of view.1405

So, look for an angle: this is the angle that is given, so we are going to use this angle right here.1409

And then, what sides of the triangle are we working with?1416

Are we working with the opposite? No, we don't have the measure of the opposite one.1421

We have the adjacent, and we have the hypotenuse.1428

Now, if I were to write out Soh-cah-toa again, just so it is easier to see,1432

the three ratios are going to be sin(x) = opposite/hypotenuse; the next one, cos(x), equals adjacent/hypotenuse;1439

and then, the last one is tan(x), equal to opposite/adjacent.1459

Which one of these three would we use?1473

We don't have the opposite; we only have the adjacent, which is this side right here1480

(because that is what we are looking for, so we have to include that one) and the hypotenuse.1486

Which one uses, from this angle's point of view, the adjacent and the hypotenuse?1492

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.1498

Now, do we have the angle measure--do we have x?1509

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.1513

Cosine of 55 equals adjacent (what is the one adjacent? That is x), over (what is the hypotenuse?) 12.1522

Now, we are going to go to our calculator, and we are going to find x.1538

Cosine of 55: now, 55 is the angle measure, so I can just punch in cos(55); cosine of 55 is .5736.1546

That equals...all of this is equal to this; that equals x/12.1568

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.1573

Then, x becomes...I just have to multiply: instead of clearing it, I can just leave that number, and then just multiply it to 12.1583

And I get 6.8829: so, this right here, this length, is 6.8829.1595

Again, to go over what we just did: cos(55)...now, why did we use cosine, and not sine and not tangent?1617

It is because we have to look at what we have.1625

And again, we don't have to use all three of them; we just have to use the one that we need,1629

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.1634

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.1641

So, you are going to use cos(x) = adjacent/hypotenuse; cos(55) =...adjacent is x; hypotenuse is 12.1656

You punch this into your calculator; you get this; then you multiply 12 to both sides, and you get the answer.1666

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.1675

I could do that, and then, if you want, you can use this angle's point of view.1687

This right here is actually going to be 35 degrees; and then, instead of using 55, you can use this one.1692

If you decide to use that one, then it is a different perspective, a different point of view.1705

So then, what would you have to use?1710

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,1713

because sin(35) is opposite over the hypotenuse; so you have options.1722

You don't have to do both; you just have to use one of them.1731

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.1733

But either way, that is an option for you, if you would like to just use that angle instead.1743

Let's do the next one: here we have...1751

And we are going to have to use this one; it is not like this one, where we have options,1756

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.1760

And then, plus, we can't subtract it from 180 (because we don't know what this angle is) to find that angle.1765

We have to use this angle; from this angle's point of view, you have to work with the side opposite and the hypotenuse.1771

So, you have opposite, and you have hypotenuse; what are you going to use?1785

With opposite and hypotenuse, you are going to have to use sine, because that uses the side opposite and the hypotenuse.1790

The other ones use adjacent and the hypotenuse (no, we don't have the adjacent);1801

this one uses opposite, which we have, but no adjacent; so we have to use the sine.1806

Then, sine...and then, what would go as the angle measure?1814

We don't have the angle measure; that is what we are looking for.1820

So, we are going to put x right there; that equals...the side opposite is 21, over...what is the hypotenuse?...25.1822

Then, from here, I am going to just go to my calculator.1835

And again, don't punch in sine of this number, because then your calculator would think that this is the angle measure.1842

But it is not; so you are going to do inverse sine; that is 2^nd and sine; you should see that sin^-1.1851

And then, 21 divided by 25...and then, you should get...1865

That automatically just gives you x, because you already used the inverse sine.1875

57.1 degrees: that is this angle measure; 57.1 degrees--that is x.1881

That is it; just remember that you have to look for an angle.1897

That way, you know from which angle's perspective you have to look to see what you have.1905

And then, from there, based on what you do have, what sides are given, and what you are looking for,1912

you are going to have to pick one of these: sine of x, cosine of x, or tangent of x.1918

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.1924

Let's do the last example: this one is a little bit tricky, but it is not difficult at all.1932

Just remember that, when you are given points like this, and you have to find the measure of angle B,1944

now, when you are given this problem, you don't know if you have to use trigonometric functions yet.1953

So, let's just plug in the points first and see what we are dealing with.1963

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.1972

So, here is the triangle; I know that this is a right angle.2001

I know that because here, this point was (-1,-5); this is -5; and this point right here was (-6,-5).2013

So then, these two points are on the same y-value right there, so then it makes a horizontal line.2028

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.2039

A horizontal and a vertical--they are perpendicular.2050

Now, what was this point? This point was A; this was B; and then, this right here is C.2055

And we want to find the measure of angle B; so then, this is the angle that we have to find.2067

Now, this is our variable; and then, let's look for side length.2076

Because here, it is perfectly horizontal or perfectly vertical, we wouldn't have to use the distance formula--we can just count.2087

Now, when it comes to BC, I am going to have to use the distance formula if I want to find BC.2097

Why?--because it is going diagonally, and you can't start counting to see the distance when it is going diagonally.2104

When it is going horizontally or vertically, then you can; you can just count the units.2113

So here, this is (-1,-5), and this one is (-6,-5); from -6 all the way to -5, this is 5 units.2120

If you have a bigger graph, you can start counting: it would just be from here, 1, 2, 3, 4, 5; so this is 5.2136

Then, this right here, the vertical--we can count that also, so here is -5: 1, 2, 3, 4, 5, 6, 7; this is 7.2146

AB is 5; AC is 7; we don't know BC.2160

Now, if you want, you can go ahead and solve for BC by using the distance formula.2165

The distance formula is x₂, the second x, minus the first x, squared, plus (y₂ - y₁)².2170

That is the distance formula; so you can just, using the coordinates for B and using the coordinates for C,2183

just plug it into this formula to find the distance of B to C.2188

But I don't think I will need it; now, let me re-draw this triangle...here is A, B, C.2194

This is what I am looking for, here; this is 5, and this is 7.2209

Now, if I am looking for this angle measure, what do I have to work with?2216

I have the side opposite, which is 7, and I have the side adjacent.2220

So, I know that I am going to use...Soh-cah-toa: from here, which one uses opposite and adjacent?2231

That would be this one right here, so we are going to use tangent.2246

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...2250

tangent of b equals opposite (what is the side opposite? It is 7), and...what is the side adjacent? 5.2260

So then, here is my equation: tan(b) = 7/5.2274

Then, you just go straight to your calculator.2279

Now, remember: don't forget; don't push tan(7/5), because that is not the angle measure; b would be the angle measure.2284

Push 2^nd, tan, to get the inverse tangent, 7/5.2290

I just close the parentheses; then, the calculator knows that I am giving them this answer, and that I want the angle measure.2299

The angle measure that it gave me was 54.46 degrees.2313

And that is it; now, when do we use Soh-cah-toa--when do we use trigonometric functions?2321

First of all, you must have a right triangle; they must be right triangles.2335

And #2: when you are dealing with angles and sides together (you are given an angle measure,2344

but you are looking for a side, using that angle measure; or given sides, you have to look for an angle measure)--2354

anything that uses a combination of angles and sides of a right triangle is when you are going to use Soh-cah-toa.2360

What if they give you two sides, and they just want you to find the other side, the missing side, the third side?2368

Well, we don't have to use Soh-cah-toa, because no angles are involved.2377

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.2381

Now, do we have to use Soh-cah-toa here--do we have to use trigonometric functions here?2397

No, because no angles are involved here; then what can we use?2401

We can use the Pythagorean theorem, remember, because 5², a², plus b², equals c².2409

Again, you are only using Soh-cah-toa when you are given sides, and then they want you to find the angle measure;2418

or given the angle measure, to find the missing side; and so on.2426

We are going to continue trigonometric functions next lesson.2431

For now, thank you for watching Educator.com.2435