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's some random right triangles that have a 41 degree angle. Of course, there are many different size 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, and 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 would be the same in all the rest of them. The 41 degree angle's between a leg and a hypotenuse. We will 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 equal, sine of 41 degrees equals opposite over hypotenuse.
The cosine equals adjacent over hypotenuse. The tangent equals 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 SOH.
Cosine is adjacent over hypotenuse; that's the CAH. 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 changed 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're 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 setup. We have a right triangle. We have an angle of ten degrees, a tiny little acute angle. And opposite that ten degree angle, the opposite side, is three centimeters. We want to find the other two lengths, for example. Well, certainly we know that the sin(10°)=opp/hyp=3/AB.
Now if we multiply both sides by AB we get AB*sin(10°) = 3, so we divide by that. And if we needed, we can 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 opp/adj, this would be 3/AC. Same thing, multiply by AC/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's 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 three, four, five triangle. That's very important to notice because the test will often expect you to recognize a three, four, five triangle.
So that missing side XZ has to equal four. Now notice that we want the tangent of angle X. From the perspective of X, three is the opposite side, and four 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 from X, three is opposite, and XZ = 4, that's the adjacent. And of course tangent is opposite of adjacent, so the opposite is YZ, the adjacent is XZ, and that is 3/4. 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, and 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/SQ. Well that's good because we know SQ. That's h/SQ, we need that h. h = 5 x tan(35 degrees). And here we can use the approximation they give us, tangent of 35 degrees is 0.7.
Well, 5 x 0.7 is 3.5, so h = 3.5. Very useful. Now that we know h, we can find the area. Of course, the area of a triangle is 1/2bh. So that's 1/2(8), which is the full base from S to R is a length of 8. 1/2(8)(3.5), 1/2 of 8 is 4.
Then for 4 x 3.5, we'll use the doubling and halving trick. 1/2 of 4 is 2. Double the 3.5 is 7. 2 x 7 is 14. That's the area. So the area is 14.
In general, for general angel, mathematicians typically use the Greek letter theta. We can use this to make general statements, true for any angle. So the sin of theta is opp/hyp. The cosine is adj/hyp, and the tangent is opp/adj. And this is the basic SOHCAHTOA pattern.
Right now, these are true when we are talking about angles inside triangles. So that means theta would have to be greater than 0 degrees and less than 90 degrees. It would 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 the equal 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 hypotenuse squared which is one.
We'll get adjacent squared divided by hypotenuse squared, while adjacent divided by hypotenuse is cosigned. An opposite divided by a hypotenuse is sine. So we get cosine squared + sine squared = 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. This is a very good one to know. Here's another practice problem. Pause the video and read this. 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 change. So we're gonna need that. So we need the cosine to relate the adjacent by hypotenuse.
Is the cosine of 35 degrees is adjacent over hypotenuse; that's PR over 25. So, PR would equal 25 x cos (35 degrees). Very good. So, we have that length; the length of that entire segment, PR. Well, PR does 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, 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 = PR. The two little segments together add up to the big segment. So that means that 3 + DR = 25 x cos(35 degrees). That's the expression we got for PR. So if we want DR, we subtract three from both sides, and that's the expression for the change in height.
And 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 sign 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 you pick any angle, say 23 degrees, a 23 degree right triangle, any 23 degree right triangle, is going to be similar to any other, and that's why all of 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.