Trigonometry, introduction to SOHCAHTOA. This lesson assumes that you are familiar with the ideas of similar triangles covered in the geometry module. If the idea of similar triangles is absolutely unfamiliar to you, it might be helpful to watch that video in the geometry module before watching the trigonometry videos. Read full transcript
Recall that if we know just two angles in one triangle are equal to two angles in the other triangle, then the two triangles must be similar. And that means that they have the same basic shape. One is just a scaled up or a scaled down version of the other. All three angles are the same, and it's the same basic shape. Once we know that the two triangles are similar, we know that all their sides are proportional.
It's very easy to show that two triangles are similar, and once we know that, we get a lot of information. All of trigonometry is based on these key pieces of information about similar triangles. Suppose we think about all the right triangles in the world with, say, a 41 degree angle.
So here are some random right triangles that have a 41 degree angle. Of course, there are many different sizes and orientations, but all have the same basic shape. All of these 41 degree right triangles are similar, because they all share a 41 degree angle, as well as a 90 degree angle. That's two angles they share in common, so they have to be similar.
This means all the sides are proportional. In other words, I could find the ratio in any one of them, and all these same ratios with the same in all the rest of them. The 41 degree angle is between a leg and a hypotenuse. We would call that leg, the leg that touches the 41 degree angle, the leg that is adjacent to that angle.
The other leg is opposite from the 41 degree angle, so we call that the opposite. So here we have the triangle with the three sides labeled, the hypotenuse, the opposite, and the adjacent. Now, the three principle ratios here are the sine ratio, sine = sin(41 degrees) = opposite over hypotenuse.
The cosine = adjacent over hypotenuse, the tangent = opposite over adjacent. Students often remember those three ratios using the mnemonic SOHCAHTOA. What is meant by SOHCAHTOA? Well, SOHCAHTOA, sine is opposite over hypotenuse, that's the S-O-H. Cosine is adjacent over hypotenuse, that's the S-A-H.
And tangent is the opposite over adjacent. So we have to remember that it's SOH-CAH-TOA. Notice that all three of these are written as functions of the angle 41 degrees, because if we change the angle, all the ratios would be different. Nevertheless, as long as we have a 41 degree right triangle, no matter the size or orientation, all these ratios will be the same.
The sine and cosine and tangent of 41 degrees, and of any other possible angle, are already stored in your calculator. You just have to make sure that your calculator is in degrees mode instead of radians mode. We'll talk more about radians in an upcoming video. Therefore, if we are given a right triangle with one known acute angle and one known length, we can always find the other two lengths.
So suppose we have this set up, we have a right triangle, we have an angle of 10 degrees, a tiny little acute angle, and opposite that 10 degree angle, the opposite side is 3 centimeters. And we want to find the other two lengths, for example. Well, certainly we know that the sine of 10 degrees is opposite over hypotenuse. So that would be 3 over side AB.
Now, if we multiply both sides by AB, we get AB times sin(10) = 3. Divide by sin(10), sin(10) is some number. So we divide by that, and if we needed, we could compute this on a calculator, sin(10) degrees is about 0.1736. 3 divided by that number is about 17.3, that's the length of the hypotenuse, AB.
We could also find side AC. We know that the tangent of 10 is opposite over adjacent. This would be 3 over AC. Same thing, multiply by AC, divide by tan(10). Now, we can find this in our calculator, tan(10) is about 0.1763, 3 divided by that number, is about 17.0.
And so we could find the two other lengths purely from the angle and the one given length. This is very powerful. Here is a practice problem. Pause the video and then we'll talk about this. Okay, the first thing to notice is what we have here is a 3-4-5 triangle.
That's very important to notice because the test will often expect you to recognize a 3-4-5 triangle. So that missing side, XZ, has to equal 4. Now, notice that we want the tangent of angle X. From the perspective of X, 3 is the opposite side, and 4 is the adjacent side, very important.
It would be very different if we were finding the tangent from Y, but from the point of view of X, 3 is opposite and XZ equals 4, that's the adjacent. And of course, tangent is opposite over adjacent, so the opposite is YZ, the adjacent is XZ, and that is 3 over 4. So it has to be answer choice E. Here's another practice problem.
Pause the video and then we'll talk about this. Okay, so we're given an angle, we're given two lengths, SQ and QR, and we're also told that the tangent of 35 degrees is approximately 0.700, and we want to know the area of the triangle. Well, we already know the base, we need the height, we need the length of PQ in order to figure out the area of the triangle.
Well, we know that the tangent of 35 degrees, that involves PQ, that's PQ over SQ. Well, that's good, because we know SQ, that's h over SQ, and we need that h. h = 5 times tangent of 35 degrees, and here we can use the approximation they give us, tangent of 35 degrees is 0.7. Well, 5 times 0.7 is 3.5, so h equals 3.5, very useful.
Now that we know h, we can find the area, of course, area of a triangle is one-half base times height. So that's one-half 8, which is the full base from S to R, is a length at 8. One-half 8 times 3.5, one-half of 8 is 4, then for 4*3.5, we'll use the doubling and halving trick.
Half of 4 is 2, double of 3.5 is 7, 2*7 is 14, that's the area. So the area is 14. In general, for a general angle, mathematicians typically use the Greek letter theta. We can use this to make general statements true for any angle. So the sine of theta is opposite over hypotenuse, the cosine is adjacent over hypotenuse, and the tangent is opposite over adjacent.
This is the basic SOHCAHTOA pattern. Right now, these are true when we're talking about angles inside triangles. So that means theta would have too be greater than 0 degrees and less than 90 degrees. We'd have to have a possible acute value inside a triangle. Right now, that's where we're gonna focus.
In this video, I'll just discuss one more important relationship that you may have to know on the test. We know, of course, from the Pythagorean theorem that the adjacent squared, plus the opposite squared, has to equal the hypotenuse squared. That's obviously true because of the Pythagorean theorem. We'll divide each term by hypotenuse squared.
On the right side, we'll get a hypotenuse squared divided by a hypotenuse squared, which is 1. We'll get adjacent squared divided by hypotenuse squared. Well, adjacent divided by hypotenuse is cosine, and opposite divided by hypotenuse is sine. So we get cosine squared plus sine squared equals 1, and this is the Pythagorean Identity.
Notice incidentally, when we square trig function, we write the square after the name of the function and before the angle. So we write it as cosine squared theta, or sine squared theta. So this is an important trig formula, and we'll return to this a few times, but this is a very good one to know. Here's another practice problem, so pause the video and read this.
And here are the expressions from which to choose, take a good look at these, and see if you can solve the problem on your own. You can pause the video and when you're ready, resume and we'll solve it together. Okay, let's think about this. We're gonna draw a right triangle with the rope as the hypotenuse, the horizontal base at the level of the tip of the prow, which is slightly above the water.
And the height up to the top of the pole, which is at P. Okay, well from the 35 degree angle at P, PR, that segment PR is the adjacent side, and that's gonna help us with the vertical chain, so we're gonna need that. So the cosine, we need the cosine to relate the adjacent to the hypotenuse. The cosine of 35 degrees is adjacent over hypotenuse, that's PR over 25, so PR would equal 25 times the cosine of 35 degrees, very good.
So we have that length, the length of that entire segment PR. Well, PR, that's not exactly the length that we're looking for. The question asks very specifically the change in level between high tide and low tide. So at high tide, the prow of the boat was at the level of D, it was at the level of the surface of the dock.
And at low tide, the prow is at the level of R and B, that horizontal line at the bottom of the triangle. So what we need, the change in level, is DR. DR is the difference between high tide and low tide. Well, we know that PD + DR = the length PR, the two little segments together add up to the big segment.
So that means that 3 + DR = 25*cos(35 degrees), that's the expression we got for PR. So if we want DR, we subtract 3 from both sides, and that's the expression for the change in height. Then we go back to the answer choices and we choose this one, answer choice C. In summary, it's good to know SOHCAHTOA, which means that the sine of theta is the opposite over hypotenuse, the cosine of theta is the adjacent over the hypotenuse.
And the tangent is the opposite over the adjacent. For any angle greater than 0 and less than 90 degrees, all the right angles with that acute angle are similar. And so all these ratios are the same for all of them. So if you pick any angle, say 23 degrees, a 23 degree right triangle, any 23 degree right triangles can be similar to any other.
And that's why all these ratios are the same. And you can find the values for these three ratios on your calculator, although the test often supplies any numbers you need.