CA Geometry: More trig

We're on problem 66. In the accompanying diagram, the measure of angle a is 32 degrees. And AC is equal to 10. Which equation could be used to find x in triangle ABC? So we want x. So let's write our SOHCAHTOA down. What are they giving us? Well x is the opposite side, this is equal to the opposite, it's opposite of our angle in question. And then this 10 side is the adjacent. So I haven't looked at the choices yet, but if we say the tangent of 32 degrees should be equal to the opposite side, which is x, over the adjacent side, which is 10. And then if we multiply both sides of this by 10, we get 10 tangent of 32 degrees is equal to x. Let's hope that's one of the choices. Yep, sure enough, C. x is equal to 10 tangent of 32 degrees. Everything in trigonometry really does boil down to SOHCAHTOA. You'll be amazed. When you actually take trigonometry as a class, you'll see that they have all these trig identities and all that. But all of those things can be just proved from SOHCAHTOA. And these are just definitions of ratios that have been very useful in the world when studying triangles. All right, 67. But you can watch, we have a whole playlist on trigonometry and we talk about all the ratios and the unit circle and all that. So if you're enjoying these problems, I encourage you to watch those videos. Or if you don't understand these problems, I encourage you even more to watch those videos. All right. The diagram shows an 8 foot ladder leaning against a wall. The ladder makes a 53 degree angle with the wall. Which is closest to the distance up the wall the ladder reaches? So this is what they care about, this distance right here. So let's see how we can figure it out. If we write SOHCAHTOA, maybe we don't have to to write that anymore. So we have this angle and what have they given us? This deals with the hypotenuse, which is 8. And it deals with this adjacent side. So what trig function deals with the adjacent side and the hypotenuse. Well, I'll write it down. SOHCAHTOA. We want to figure out adjacent, we have the hypotenuse. What trig function deals with it? Well, cosine is adjacent over hypotenuse. So the cosine of 53 degrees is equal to the adjacent side of this triangle, of this right triangle. So that's A. And you should only be applying these, right now, with right triangles. Later on we'll learn, in trigonometry how trig functions are useful for any triangle. Because obviously, only a right triangle has a hypotenuse. So these h's wouldn't be too meaningful if you're not dealing with a right triangle. Back to the problem. Cosine of 53, it equals the adjacent side over the hypotenuse, which is 8. I can write x there, x is the adjacent side. So we multiply both sides of this by 8, we get 8 times the cosine of 53 degrees is equal to x. And they give us a little chart. They tell us that the cosine of 53 degrees is approximately equal to 0.6. So we get 8 times 0.6 is equal to x. 8 times 0.6 is 4.8. So x is equal to 4.8. And that is choice B. Problem 68. Triangle JKL is shown below. Fair enough. Which equation should be used to find the length of JK? This is JK. All right so we have this angle here. JK is the opposite side. We want to figure out the opposite side. We have the hypotenuse. So what trig function deals with opposite and hypotenuse? Well I could just write SOH down. I'm don't even have to write the whole SOHCAHTOA. Because sine is equal to the opposite side over the hypotenuse. So in this case, sine of 24 degrees is equal to the opposite side, which is side JK, over the hypotenuse. This is the hypotenuse right here, 28. The longest side of the right triangle, or the side that's opposite the 90 degree angle. And so maybe we're done. So sine of 24 degrees is equal to JK over 28. That's choice A. Problem 69. These are kind of fun, aren't they? What is the approximate height in feet of the tree in the figure below? Now you'll see why trigonometry is useful, because I'm sure you often wonder how tall is that tree? And it might be hard to climb it. And now you can use trigonometry. If you just have a way to measure the angle between your line of sight and between you and the top of the tree. You've probably seen those surveyors, actually they use things like this. And you can figure out the heights of things far away if you know how far away you are from it. But anyway, let's work on this problem. What is the approximate height in feet of the tree in the figure below? OK, so this is 50. So let's think about it. What do they tell us? This is 90, this is 50. So the approximate height in feet. OK, so this is the opposite side of this angle. So let's call this h. So what deals with the opposite side? Actually I should use h, let's call that O. O is the height of the tree because it's opposite the 50 angle. So what deals with the opposite and the hypotenuse? So once again, SOHCAHTOA. We just have to look at the SOHA part. Sine is equal to opposite over hypotenuse. So the sine of 50 degrees is equal to the opposite side of this triangle, that's the height of the tree. So I'll say O for the height of the tree. Over the hypotenuse, the hypotenuse is 100 feet. Multiply both sides by 100. You get 100 sine of 50 is equal to this opposite side, which is equal to the height of the tree. Sine of 50 is 0.76. So 0.76 times 100 is equal to O. That 76 is equal to O, or the height of the tree. Oh, they said 0.766. So let's put another 6. That becomes 76.6. And that's choice B. All right. Problem 70. These go fast. If a is equal to 3 roots of 3 in the triangle below. I think you're going to learn a little bit about 30, 60, 90 triangles now. What is the value of B? OK, maybe they assume that you already do know something about 30, 60, 90 triangles. But anyway, if a is equal to 3 root 3, what is B? And I'm debating on what I should assume that you already know. But let's think about it a little bit. Let me see how I can do this without resorting to just the definition of 30, 60, 90 triangles. I'll prove it for you. Although, later you can memorize. Because when I keep saying 30, 60, 90 triangles, we know that this right here is 60 degrees. Because 30, plus 60, plus 90 is equal to 180 degrees. So that's why we could we keep calling this a 30, 60, 90 triangle. So how do we figure out this side, if we just know that side? And we know this is 30 degrees. So what I'm going to do is, I'm going to try to redraw this same triangle but I'm going to flip it over. And this is actually the proof of figuring out the measures of a 30, 60, 90 triangle. So maybe I'll do it in a different color. And I have some time so I'll do it. Let's see. So I'll do it like that. I have to bring another line down, something like that. And then I'll have to draw another line more like that. I think you get the idea. And then bring a line across like that. And actually, I'm going to draw another line straight down like that. OK, let's see what we can do with this. Maybe there's an easier way to do it. But this is just what my brain is thinking of right now. Let's think about it a little bit. In this drawing that I've done, this triangle is just a mirror image of that one. So this right here, this is also 60. This is 60, this is 60, what's this angle going to be? Well they're all collectively supplements of each other. If you go all the way from there to there, you have 180 degrees. So this big angle right here has got to be 60 degrees. This angle right here, it's a complement of 30. So this is 60 degrees. And then this is 60 degrees. So this is an equilateral triangle. All the sides are going to be equal. So that side is equal to that side, which is equal to this side. Now let me ask another question. What is this side right here? Let me just draw the shorter part. And once you memorize 30, 60, 90 triangles, you don't have to go through all this. But it's good to be able to reprove it. So what's this length right here? Well if we look right up there, that's length a. This was just a mirror image, so this is also length a. This whole base of the equilateral triangle is 2a. Well it's an equilateral triangle so all the sides are the same, so this is 2a and this is 2a. And just like that we were able to figure out the hypotenuse. And they want to know what the b is. Once we know two sides of a right triangle, it's very easy to figure out the third side. So we know that a squared plus b squared is equal to c squared. Let me write that down. Now what's a squared? a is 3 roots of 3. So a squared would be, let me write it down, 3 roots of 3 squared plus b squared is equal to, let me do it in another color. What's c? c, we just figured out, is 2 times a. So it's 6 roots of 3 squared. That's what we did all this stuff for, to figure out that this length is twice that length. Fair enough. Now let's simplify. So if we take 3 roots of 3 to the second power, that's the same thing as 3 squared times square root of 3 squared. So that's 9 times 3 plus b squared is equal to 36 times 3. And so that's 27 plus b squared. 36 times 3 is 108. Subtract 27 from both sides. b squared is equal to 81. b is equal to 9. So that is choice a. Anyway, you should watch the videos I've done in the trigonometry playlist on 30, 60, 90 triangles if you want to be able to do this faster. But I think that was useful. Because you've actually seem how you can figure out the sides of a 30, 60, 90 triangle without having memorized it ahead of time. Anyway, see you in the next video.