WEBVTT mathematics/trigonometry/murray
00:00:00.000 --> 00:00:05.000
Hi, these are the trigonometry lectures for educator.com.
00:00:05.000 --> 00:00:09.000
Today, we're going to talk about trigonometry in triangles that have a right angle.
00:00:09.000 --> 00:00:11.000
These are called right triangles.
00:00:11.000 --> 00:00:17.000
The master formula for right triangles, we've seen it before it's the SOH CAH TOA.
00:00:17.000 --> 00:00:22.000
That's the word that I remember to know that the sin(θ) ...
00:00:22.000 --> 00:00:26.000
Let me draw a right triangle here.
00:00:26.000 --> 00:00:42.000
If θ is one of the small angles, not the right angle, then the sin(θ) is equal to the length of the opposite side over the length of the hypotenuse.
00:00:42.000 --> 00:00:50.000
Cos(θ) is equal to the adjacent side over the hypotenuse.
00:00:50.000 --> 00:00:53.000
The tan(θ) is equal to the opposite side over the adjacent side.
00:00:53.000 --> 00:01:03.000
For shorthand, sine is equal to opposite over hypotenuse, cosine is equal to adjacent over hypotenuse, tangent is equal to opposite over adjacent.
00:01:03.000 --> 00:01:11.000
It's probably worth saying SOH CAH TOA often enough until it sticks into your memory, because it really is useful for remembering these things.
00:01:11.000 --> 00:01:23.000
If you have a hard time remembering that, the little mnemonic that is also helpful for some students is Some Old Horse Caught Another Horse Taking Oats Away.
00:01:23.000 --> 00:01:25.000
That spells out SOH CAH TOA for you.
00:01:25.000 --> 00:01:33.000
One key thing to remember here is that SOH CAH TOA only works in right triangles.
00:01:33.000 --> 00:01:36.000
You have to have one angle being a right angle.
00:01:36.000 --> 00:01:56.000
If you have a triangle that is not a right triangle, if you don't have a right angle, don't use SOH CAH TOA because it's not valid in triangles that don't have right angles.
00:01:56.000 --> 00:02:02.000
That's if you have no right angle.
00:02:02.000 --> 00:02:08.000
We're going to learn in the next lectures on educator.com, we'll learn about the law of sines and the law of cosines.
00:02:08.000 --> 00:02:18.000
Those work in any triangle where you don't need a right angle, but when you have a right angle, it's definitely easier and better and quicker to use SOH CAH TOA.
00:02:18.000 --> 00:02:20.000
Let's try that out with some actual triangles.
00:02:20.000 --> 00:02:29.000
On the first example, we have a right triangle with short sides of length 3 and 4, and we want to find all the angles in the triangle.
00:02:29.000 --> 00:02:34.000
Let me draw a triangle here.
00:02:34.000 --> 00:02:38.000
We're told that the short sides have length 3 and 4.
00:02:38.000 --> 00:02:42.000
Of course, one angle is a right angle, so I don't need to worry about that.
00:02:42.000 --> 00:02:46.000
I'll call these angles θ, and I'll call this one φ.
00:02:46.000 --> 00:02:52.000
First thing we're going to need to know is what the hypotenuse of this triangle is.
00:02:52.000 --> 00:03:04.000
h²=3²+4², Pythagorean theorem there, which is 9+16, which is 25.
00:03:04.000 --> 00:03:07.000
So, h is the square root of 25, h is 5.
00:03:07.000 --> 00:03:11.000
Let me draw that in there.
00:03:11.000 --> 00:03:17.000
Now, I want to figure out what θ and φ are.
00:03:17.000 --> 00:03:20.000
I'm going to use SOH CAH TOA.
00:03:20.000 --> 00:03:25.000
Let me write that down there for reference, SOH CAH TOA.
00:03:25.000 --> 00:03:34.000
I'm going to figure out what θ is by using the SOH part of the SOH CAH TOA.
00:03:34.000 --> 00:03:45.000
Sin(θ) is equal to the opposite over the hypotenuse.
00:03:45.000 --> 00:03:51.000
The opposite angle to θ is 3 and the hypotenuse is 5.
00:03:51.000 --> 00:03:58.000
So, θ=arcsin(3/5).
00:03:58.000 --> 00:04:01.000
I'm going to work that out on the calculator.
00:04:01.000 --> 00:04:11.000
My calculator has an arcsine button, it actually writes it as sin^-1, which I don't like that notation because makes it seem like a power.
00:04:11.000 --> 00:04:17.000
In any case, I'm going to use inverse sine of 3/5.
00:04:17.000 --> 00:04:30.000
There's a very important step here that many students get confused about which is that, if you're looking for an answer in terms of degrees, which in real world measurement, it sometimes easier to use degrees than radians.
00:04:30.000 --> 00:04:34.000
You have to set your calculator to degree mode.
00:04:34.000 --> 00:04:38.000
Most calculators have a degree mode and a radian mode.
00:04:38.000 --> 00:04:44.000
In fact, all calculators that do trigonometric functions have a degree mode and a radian mode.
00:04:44.000 --> 00:04:47.000
The default is probably radian mode.
00:04:47.000 --> 00:04:57.000
If your calculator is set in radian mode and you try to do something like arcsin(3/5), you'll get an answer that doesn't look right.
00:04:57.000 --> 00:05:04.000
You may be confused if you're checking your answers somewhere, it may not agree with what the correct answer it.
00:05:04.000 --> 00:05:11.000
What you have to do is set up your calculator in degree mode, if you want an answer in degrees.
00:05:11.000 --> 00:05:15.000
My calculator is a Texas Instruments.
00:05:15.000 --> 00:05:23.000
It's got a mode button, I just scroll down, it say's RAD and DEG to convert it from radians to degrees.
00:05:23.000 --> 00:05:30.000
I'm going to convert it into degree mode and then I'll get an answer in terms of degrees.
00:05:30.000 --> 00:05:38.000
That's a step that many students forget and they get kind of confused when they get an answer which is in terms of radians, but it doesn't agree with the degree answer they were looking for.
00:05:38.000 --> 00:05:42.000
Now, I've got my calculator set in degree mode.
00:05:42.000 --> 00:05:48.000
I'll do the arcsin(3/5) which is 0.6.
00:05:48.000 --> 00:05:58.000
It tells me that that is approximately equal to 36.9 degrees.
00:05:58.000 --> 00:05:59.000
I found one of the angles in the triangle.
00:05:59.000 --> 00:06:02.000
Of course, another one is a right angle, so it's 90 degrees.
00:06:02.000 --> 00:06:09.000
For φ, I think for φ, I'm going to practice the cosine part of the SOH CAH TOA.
00:06:09.000 --> 00:06:18.000
I know that cos(φ) is equal to adjacent over hypotenuse.
00:06:18.000 --> 00:06:30.000
That's the adjacent side to φ, so φ is over here, it's adjacent side is 3, hypotenuse is still 5.
00:06:30.000 --> 00:06:36.000
So, φ=arccos(3/5).
00:06:36.000 --> 00:06:44.000
Again, I'll do that on my calculator.
00:06:44.000 --> 00:06:52.000
The calculator tells me that that's 53.1 degrees, approximately equal to 53.1 degrees.
00:06:52.000 --> 00:06:56.000
Now, there's a little check here you can do to make sure that you work this out correctly.
00:06:56.000 --> 00:07:01.000
We know that the angles of a triangle add up to 180 degrees.
00:07:01.000 --> 00:07:19.000
If we check here, 36.9+53.1 plus the last angle was a 90-degree angle, a right angle, if you add those up, 36.9+53.1 is 90 plus 90, you get 180 degrees.
00:07:19.000 --> 00:07:23.000
That tells me that my answers are right.
00:07:23.000 --> 00:07:26.000
The key formula here to remember is SOH CAH TOA.
00:07:26.000 --> 00:07:40.000
Everything comes down to drawing the angles in the triangle, and just using sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent.
00:07:40.000 --> 00:07:52.000
Now, we're given a right triangle and we're told that one angle measures 40 degrees.
00:07:52.000 --> 00:08:00.000
Let me call that angle right here, that would be 40 degrees.
00:08:00.000 --> 00:08:06.000
The opposite side has length 6.
00:08:06.000 --> 00:08:11.000
I want to find the lengths of all the sides in the triangle.
00:08:11.000 --> 00:08:18.000
Of course, finding the angles is no big deal because one side is a right angle, we're told that it is a right triangle.
00:08:18.000 --> 00:08:22.000
I can find the other angle just by subtracting from 180.
00:08:22.000 --> 00:08:31.000
In fact, 180-40 is 140, minus 90, is 50.
00:08:31.000 --> 00:08:33.000
I know that other angle is 50.
00:08:33.000 --> 00:08:36.000
The challenge here is to find the lengths.
00:08:36.000 --> 00:08:41.000
We're going to use SOH CAH TOA.
00:08:41.000 --> 00:08:46.000
I'm going to apply SOH CAH TOA to 40.
00:08:46.000 --> 00:08:55.000
I know that sin(40) is equal to 6 over the hypotenuse, because that's opposite over hypotenuse.
00:08:55.000 --> 00:09:05.000
The hypotenuse, if I solve this, that's equal to 6/sin(40).
00:09:05.000 --> 00:09:11.000
Remember to convert your calculator to degree mode before you do this kind of calculation.
00:09:11.000 --> 00:09:21.000
6/sin(40) is approximately equal to 9.3.
00:09:21.000 --> 00:09:27.000
That tells us the length of the hypotenuse, 9.3.
00:09:27.000 --> 00:09:32.000
Now I'd like to find the length of the other sides.
00:09:32.000 --> 00:09:34.000
I'm going to use the cosine part of SOH CAH TOA.
00:09:34.000 --> 00:09:45.000
I know that cos(40) is equal to the adjacent over the hypotenuse.
00:09:45.000 --> 00:09:52.000
If I solve that for the adjacent side, that's equal to hypotenuse times the cos(40).
00:09:52.000 --> 00:09:56.000
I know the hypotenuse now.
00:09:56.000 --> 00:10:01.000
If I multiply that by cos(40).
00:10:01.000 --> 00:10:11.000
What I get is approximately 7.2 for my adjacent side.
00:10:11.000 --> 00:10:18.000
The sides of my triangle are 6, 7.2 and 9.3.
00:10:18.000 --> 00:10:20.000
Again, there's an easy way to check that.
00:10:20.000 --> 00:10:23.000
We'll check that using the Pythagorean theorem.
00:10:23.000 --> 00:10:56.000
I want to check that 6²+7.2² gives me 87.8, which is approximately equal to 9.3².
00:10:56.000 --> 00:11:02.000
I know that I got those side lengths right.
00:11:02.000 --> 00:11:08.000
The third example here, we have the lengths of the two short sides of a right triangle, are in a 5:2 ratio.
00:11:08.000 --> 00:11:16.000
Let me draw that out.
00:11:16.000 --> 00:11:19.000
We're given that it's a right triangle.
00:11:19.000 --> 00:11:23.000
We've got lengths in a 5:2 ratio.
00:11:23.000 --> 00:11:35.000
I don't actually know that these lengths are 5 and 2, but here's the thing, if I expand this triangle proportionately, it won't change the angles.
00:11:35.000 --> 00:11:46.000
If I blow this up to a similar triangle that actually has side lengths of 5 and 2, I'll get a similar triangle with the same angles.
00:11:46.000 --> 00:11:51.000
I can just assume that this triangle has actually side lengths of 5 and 2.
00:11:51.000 --> 00:11:55.000
I want to find all the angles in the triangle.
00:11:55.000 --> 00:11:59.000
I'll label that one as θ, that one is φ.
00:11:59.000 --> 00:12:02.000
I know that sooner or later, I'm going to need the hypotenuse of the triangle, I'll go ahead and find that now.
00:12:02.000 --> 00:12:18.000
h²=2²+5², that's equal to 4+25 which is 29.
00:12:18.000 --> 00:12:27.000
My hypotenuse is the square root of 29.
00:12:27.000 --> 00:12:37.000
I'd like to practice all parts of SOH CAH TOA, and we've used sine and cosine in the previous problems.
00:12:37.000 --> 00:12:40.000
I'm going to use the tangent part.
00:12:40.000 --> 00:12:45.000
Remember, tangent is opposite over adjacent.
00:12:45.000 --> 00:12:52.000
Tan(θ) there, the opposite side is 5, and the adjacent side has length 2.
00:12:52.000 --> 00:13:03.000
I'm going to find arctan(5/2), θ is arctan(5/2).
00:13:03.000 --> 00:13:20.000
Remember that you want to have your calculator in degree mode here, because if you have your calculator in radian mode, you'll get an answer in radian which would look very different from any answer in degrees that you were expecting.
00:13:20.000 --> 00:13:35.000
I calculate arctan(5/2), and it tells me that that is approximately equal to 68.2 degrees.
00:13:35.000 --> 00:13:39.000
Now, I found θ, I've got to find φ now.
00:13:39.000 --> 00:13:46.000
I could find φ just by subtracting θ from 90 degrees, because I know θ+φ adds to 90 degrees.
00:13:46.000 --> 00:13:53.000
Remember, one angle of the triangle is already 90 degrees, the other two must add up to 90 degrees.
00:13:53.000 --> 00:13:59.000
I could just find the other angle by subtracting but I want to avoid that.
00:13:59.000 --> 00:14:01.000
I want to practice using my SOH CAH TOA rules.
00:14:01.000 --> 00:14:08.000
The other reason is if I find it using some other method, then I can add them together at the end and use that to check my work.
00:14:08.000 --> 00:14:13.000
I'm going to try using a SOH CAH TOA rule.
00:14:13.000 --> 00:14:14.000
I'm going to find it using angle φ.
00:14:14.000 --> 00:14:26.000
Looking at φ, I know that sin(φ) is equal to the opposite over hypotenuse.
00:14:26.000 --> 00:14:32.000
Sin(φ) is equal to the opposite over hypotenuse.
00:14:32.000 --> 00:14:41.000
The opposite side to φ is 2, and the hypotenuse is the square root of 29.
00:14:41.000 --> 00:14:46.000
φ is equal to the arcsine of 2 over the square root of 29.
00:14:46.000 --> 00:14:51.000
That's definitely something I want to put into my calculator.
00:14:51.000 --> 00:15:11.000
I'll figure out inverse sine of 2 divided by square root of 29 ...
00:15:11.000 --> 00:15:17.000
The inverse sine of 2 divided by square root of 29 ...
00:15:17.000 --> 00:15:25.000
It tells me that that's approximately equal to 21.8 degrees.
00:15:25.000 --> 00:15:28.000
That's what I got using SOH CAH TOA.
00:15:28.000 --> 00:15:37.000
Again, I'm going to check it by checking that the three angles of this triangle add up to a 180 degrees.
00:15:37.000 --> 00:15:51.000
I've got 68.2 degrees plus 21.8 degrees, those add up to 90, plus the last 90-degree angle, the right angle.
00:15:51.000 --> 00:15:54.000
Those do add up to 180 degrees.
00:15:54.000 --> 00:15:58.000
That tells me that my work must probably be right.
00:15:58.000 --> 00:16:08.000
That came back to looking at SOH CAH TOA, and figuring out what the angles were based on the SOH CAH TOA.
00:16:08.000 --> 00:16:14.000
We know that tangent is opposite over adjacent, and we also used that sine is opposite over hypotenuse.
00:16:14.000 --> 00:25:43.000
We'll try some more examples later.