Trigonometry - Solving a Right Triangle Given Two Sides

INSTRUCTOR: Hello again, Grade Elevens.

I'm just going to run you through one more example of solving a triangle.

The difference with this one is that we are given the lengths of two sides,

and in this case, we have no angles.

I would probably say that most of you,

the first thing that you're going to want to do is to find side y here,

and the way that you're going to do that probably is to use the Pythagorean theorem.

That's great. It would certainly work.

I'm not going to use the Pythagorean theorem because for me,

I think that trig works a little bit better.

Well, it doesn't work better, it works the same,

but I do think it's a little bit quicker.

First, I have to find the angles.

I have no choice. I have to use trig at least once.

I'm going to find angle W first.

What I have to do to do this is recognize that W becomes my reference angle,

and now I have to take a look at what information I have.

I'm looking for the angle, which means I need to have two sides.

I come across from W and I recognize that I have the opposite side.

I obviously don't have the hypotenuse yet,

so that means I must also have the adjacent.

Now, according to SOHCAHTOA,

I need to use TOA,

which is tan, of course,

because I have O and A.

That means that for me,

tan of my angle equals,

please make sure you're doing that equal sign.

Otherwise, what you're writing doesn't actually make any sense.

You also need to make sure you include

this W because tan on its own doesn't mean anything.

What this tells us is that tan of this angle,

angle W is equal to 21.2 divided by 43.5.

This tells us the ratio of these sides for this angle,

and whatever this ratio is,

there's only one possible angle that would fit.

We're going to go ahead and solve that.

Tan W equals,

now, all we're doing here is dividing.

I go ahead and divide,

and I get 0.487356321,

but I don't need all those decimal places.

The rule that I've given you guys for

this unit is to always round to four decimal places.

The way that we do that, 1, 2, 3,

4, and I look at the fifth decimal place.

In this case, it's a five, and the rule is,

if this is five or greater,

it bumps up the one that came before it.

This becomes a four,

and we can ignore the rest of those.

I'm just going to rewrite this so it's a little bit clear.

Tan W equals 0.4874.

Now, I can't just use

the regular old tan button because I can only do that when I know the angle.

When I'm trying to find the angle,

I end up with something like this.

Angle W equals inverse tan.

Remember you hit the second button on your calculator

or the shift button depending on how your calculator works.

We're doing inverse tan of 0.4874.

I go ahead and I type that in and I get the W

equals 25.98 degrees or 26 degrees.

By the way, the reason that color is changing is because

I'm pausing it to use my calculator,

and then when I turn it back on,

it for some reason, puts something in black.

You can just ignore that black and assume it's the same color.

There's angle W. To find angle X, well, that's easy.

Angle X, we know the angles all add up to 180,

so it's 180-90,

and then we subtract 26 degrees.

180 minus 90 equals 90, 90 minus 20 is 70,

take off another six and I believe it is 64 degrees.

There we go. There's angle X.

Now, I did that in my head,

so I'm not really confident with that answer,

so I'm not going to use that in my next calculation.

I also already mentioned that I'm not going to use

the Pythagorean theorem because I'd like to practice my trig a little bit more.

I'm going to go ahead and use 26 degrees because I'm fairly confident in this answer.

Now, some of you might want to use

the Pythagorean theorem because you know that these numbers are correct,

and if you use the Pythagorean theorem,

you'll be certainly confident that your answer is correct, but I'm not going to.

I'm going to use angle W as my reference angle,

and the first thing I identify is that when I'm trying to find side Y,

I'm looking for the hypotenuse.

I know that I can either use sine because the hypotenuse is in there,

or I can use cos.

I can't use tan because the hypotenuse isn't involved,

so it's not going to give me the hypotenuse.

Now what I find is with angle W,

I can either use 21.2,

which is the opposite, or I can use 43.5, which is the adjacent.

If I use 21.2,

which is the opposite, I would use sine.

If I use 43.5,

which is the adjacent, I would use cos.

Either way, I'll get the exact same answer.

I haven't used cos yet, I don't think,

so I'm going to use the adjacent side as well.

A and H means I'm going to use cos,

the adjacent always goes on top.

Cos of my angle,

which I decided was 26 degrees is equal to 43.5 over y.

Now, what you'll notice is that in all three of these examples that I've done so far,

or I guess both videos,

my variable's been on the bottom, and that's the hard case.

The easy case is when the variable's on top.

Just remember, if your variable,

that's the side you're solving for is on top,

you're just going to multiply by what's underneath.

It's only in these cases where the variable's on the bottom that you have to switch.

Something that's helpful, if you look at SOHCAHTOA,

if you're solving for the hypotenuse,

it's on the bottom here and it's on the bottom here.

Whenever you're solving for the hypotenuse,

it's probably going to be on the bottom.

If you're ever solving for the opposite,

you can see it's on top, it's not involved, and it's on top.

If you're solving for the opposite,

it's always going to be on top, so it's always going to be the easier calculation.

And if you're solving for the adjacent,

it's either going to be on top with cos or it's going to be on the bottom with tan.

If you like it being on top,

use cos to find the adjacent,

assuming, of course, that you have the hypotenuse.

Let's finish this up.

Again, we know that these two things switch,

so we get y equals 43.5 divided by cos 26.

We're solving for y,

which is the hypotenuse,

so we know that our answer has to be bigger than 43.5 and bigger than 21.2.

I go ahead and I solve it,

and I get 48.4,

which is bigger than 43.5.

I think that answer is probably correct. So 48.4 meters.

Now I have solved this triangle.

I have all the angles.

I had two of the sides,

and I found the third one.

That is how I would go about solving this triangle,

but as I mentioned in class,

there are several ways to solve any of these,

and the way that you do it is totally up to you.

Good luck on the assignment, and I'll see you on Monday.