WEBVTT 00:00:05.350 --> 00:00:10.854 We're going to take a look at solving right triangles. To solve a right triangle means to find the measure 00:00:10.856 --> 00:00:18.854 of each of the acute angles and the length of each of the three sides. It's a good idea to organize your 00:00:18.856 --> 00:00:26.632 information from the beginning. There are three sides. We need to know the length of each. c is the 00:00:26.634 --> 00:00:36.972 hypotenuse, 10, which happens to be given this time. Then we have three angles. Angle C is the 90 degree 00:00:36.974 --> 00:00:50.218 angle opposite the hypotenuse. B is given to be 35 degrees, and A we'll need to find. Now finding angle A is 00:00:50.220 --> 00:00:55.717 the easiest thing we can do in this problem because we know that the sum of the three angles in a triangle 00:00:55.719 --> 00:01:10.971 is 180 degrees. So A is 180 minus the 35, minus the 90 degrees, and if we subtract this we should come up 00:01:10.973 --> 00:01:26.273 with 55 degrees. Ok, so what are we going to do next? We need at least one more side before we can 00:01:26.275 --> 00:01:31.017 think about the Pythagorean theorem or anything, so we're going to have to use trig. We'd like to use one 00:01:31.019 --> 00:01:41.767 of those six trig ratios that we've learned. We have some choices. Let's find b first. [Side] b is the side opposite 00:01:41.769 --> 00:01:50.703 35 degrees, so which ratio might I use? It kind of depends on what else I know. I'd like to know the 00:01:50.705 --> 00:02:01.955 opposite side. I do know the hypotenuse. So now we know which ratio to use, right? The sine of 35 degrees 00:02:01.957 --> 00:02:19.500 is the opposite side, b, over the hypotenuse. We're going to solve for b. Multiply both sides by 10. 00:02:19.502 --> 00:02:31.410 So we need our calculator. Make sure it's in degree mode, and we want to multiply 10 times the sine of 00:02:31.412 --> 00:02:47.343 35 degrees. That is 5.735. We're going to round to one decimal place, so this is approximately 5.7, and this is b. 00:02:47.345 --> 00:02:55.351 We have one more piece of information to find, and that is a. We could use the Pythagorean theorem 00:02:55.353 --> 00:03:02.105 to find it. We could also use one of our other trig ratios to figure it out. [Side] a is the adjacent side to the 35 00:03:02.107 --> 00:03:09.853 degrees, and we know the hypotenuse is exactly 10. So we're going to go ahead and use the cosine of 35 00:03:09.855 --> 00:03:28.354 degrees to find a. So it's going to be a over 10. Multiply both sides by 10 again. Use your calculator to 00:03:28.356 --> 00:03:42.604 approximate that to one decimal place. This is going to be approximately 8.2 rounded to one decimal place. 00:03:42.606 --> 00:03:45.606 Let's try one more. 00:03:47.350 --> 00:03:53.583 In this right triangle, we're given two sides, and we'd like to solve that right triangle. So here we need to 00:03:53.585 --> 00:04:18.103 find the third side and the two acute angles. Let's start by organizing our information. So a is 5, b is 6, 00:04:18.105 --> 00:04:24.868 c we don't know yet, and now we have to worry about our angles. We don't know A or B at this point, and 00:04:24.870 --> 00:04:36.627 we know that C is our right angle. We can find the hypotenuse, c, using the Pythagorean theorem, so let's 00:04:36.629 --> 00:04:54.878 do that first. [Side] c squared has to equal 5 squared plus 6 squared. So c is exactly equal to the square root of 61. 00:04:54.880 --> 00:05:14.127 We use our calculator to approximate that to one decimal place, and it's about 7.8. Now for the angles we'll 00:05:14.129 --> 00:05:22.172 have to use some of our trig ratios. So let's look at angle A. We know the opposite side to angle A is 5 and 00:05:22.174 --> 00:05:38.426 the adjacent side is 6, so that's our tangent ratio. Tangent of A is 5 over 6, so we want to use the inverse 00:05:38.428 --> 00:05:53.176 tangent on our calculator to approximate A. Inverse tangent is usually that second function on your tangent 00:05:53.178 --> 00:06:13.672 button. This is about 39.8 degrees. You need to make sure your calculator is in degree mode for that one. 00:06:13.674 --> 00:06:19.480 Ok, the only thing left is angle B. It's one of the acute angles in the right triangle. We know that the sum 00:06:19.482 --> 00:06:26.985 of all those angles is 180 degrees, so that means the sum of these acute angles is 90 degrees. 00:06:26.987 --> 00:06:38.990 The easiest way to find B then is to take 90 minus the 39.8 degrees you just found for angle A, 00:06:38.992 --> 00:06:49.700 and we find out then that B is 50.2 degrees. 00:06:51.736 --> 00:07:01.020 Solving right triangles has many applications. Here's one. A hawk is sitting on top of a 34 foot pole spots 00:07:01.022 --> 00:07:07.019 a mouse on the ground at an angle of depression of 28 degrees. Find the line of sight distance between 00:07:07.021 --> 00:07:14.519 the hawk and the mouse. So clearly before we start we need to draw some sort of a picture. Here's our 00:07:14.521 --> 00:07:23.519 34 foot tall pole that's on the ground. We're going to assume the ground is flat, and right at the top of the 00:07:23.521 --> 00:07:34.049 pole, there's our hawk, and he's looking, and he finds a mouse. The mouse is down here. The question 00:07:34.051 --> 00:07:38.797 asks us to find the line of sight distance between the hawk and the mouse. So we want to know how far 00:07:38.799 --> 00:07:45.052 is it for the hawk to get to the mouse. Ok, well we have one other piece of information we need to place 00:07:45.054 --> 00:07:53.825 on our picture. The angle of depression from the hawk looking down to the ground is 28 degrees. An angle 00:07:53.827 --> 00:08:03.325 of depression is relative to the horizontal, and it's depression. It's going down. So this is our angle of 00:08:03.327 --> 00:08:10.840 28 degrees that we're given. This dotted line, the diagonal one here, that is the line of sight distance between 00:08:10.842 --> 00:08:19.087 the hawk and the mouse, and that is what we want to find. So let's call it x. Where's the right triangle? 00:08:19.089 --> 00:08:27.109 Well we have to assume that the pole is perpendicular to the ground, so there we go. Ok, well there's one 00:08:27.111 --> 00:08:33.860 other problem. Notice that the 28 degrees is not part of the right triangle that I would like to solve, so we're 00:08:33.862 --> 00:08:42.361 going to do a little geometry. If this is a right angle, then so is this one, so that means that the 28 degrees 00:08:42.363 --> 00:08:52.866 and alpha have to have a sum of 90 degrees. So the first thing I'm going to do is subtract 90 minus 28, 00:08:52.868 --> 00:09:05.895 which is 62 degrees, so that's alpha. I can just stop there because alpha is in the right triangle. It's one 00:09:05.897 --> 00:09:14.393 of my acute angles, and I can use that one, I think, to help me figure out x. So x is the hypotenuse of the 00:09:14.395 --> 00:09:23.915 right triangle, and I know this leg. This leg is adjacent to angle alpha, and that should trigger cosine. 00:09:23.917 --> 00:09:33.175 Cosine of this angle is the adjacent side over the hypotenuse. So let's write this down. The cosine of 62 degrees 00:09:33.177 --> 00:09:50.673 is 34 over x. We know how to solve this, right? Multiply both sides by x. Divide both sides by cosine 62 00:09:50.675 --> 00:10:10.672 degrees, and we take out our calculator and approximate this to one decimal place as requested. 00:10:10.674 --> 00:10:21.425 So about 72.4 feet is the line of sight distance from the hawk to the mouse. 00:10:23.673 --> 00:10:29.729 This next example is a little bit more complicated. We have to use two right triangles to find a missing 00:10:29.731 --> 00:10:36.727 piece of information. A monument is spotted at the edge of a cliff. At a distance of 300 feet from the base 00:10:36.729 --> 00:10:42.729 of the cliff, the angle of elevation to the bottom of the monument is 37.8 degrees, and the angle of elevation 00:10:42.731 --> 00:10:49.225 to the top of the monument is 42 degrees. Find the height of the monument rounded to one decimal place. 00:10:49.227 --> 00:10:57.008 So we need to draw a picture. We're going to draw a simple picture. Here's our cliff, and we're going to put 00:10:57.010 --> 00:11:06.017 a monument of some person on the top of the cliff. Now imagine that you're standing over here, 300 feet 00:11:06.019 --> 00:11:21.015 away from the cliff, and you look up to the bottom of the monument. We were told the angle of elevation to 00:11:21.017 --> 00:11:32.515 the bottom of the monument is 37.8 degrees. The angle of elevation to the top of the monument, from that 00:11:32.517 --> 00:11:49.786 same place, is 42 degrees. We're going to assume the cliff is perpendicular to the ground, so we have a right 00:11:49.788 --> 00:12:02.070 triangle here. We were told we were standing 300 feet away, so let's label 300 feet. This is really all we know, 00:12:02.072 --> 00:12:09.072 but we do have some unknowns that will be helpful to us. We're asked to figure out the height of the monument, 00:12:09.072 --> 00:12:24.600 so the height, h, is going to be for the height of the monument, and then x is going to be the height of the cliff. 00:12:24.602 --> 00:12:31.848 We have two right triangles. We know one acute angle. Look at the 300 feet. That is the adjacent side to the 00:12:31.850 --> 00:12:42.611 37.8 degrees, and it's also the adjacent side, in the bigger triangle, to the 42 degrees. Let's concentrate on one 00:12:42.613 --> 00:12:52.626 of those triangles to begin with. Let's concentrate on the smaller one. What side is opposite 37.8 degrees? 00:12:52.628 --> 00:13:00.652 That's our x that we don't know, and the adjacent side is 300. So we're going to use tangent, opposite 00:13:00.654 --> 00:13:17.127 over adjacent. Tangent of 37.8 degrees is x over 300. Now let's look at the bigger triangle. This one has an 00:13:17.129 --> 00:13:25.155 angle of elevation of 42 degrees. The adjacent side is still 300, and the opposite side is all of this, the cliff 00:13:25.157 --> 00:13:36.907 plus the monument, or x plus h. We can write another equation. Tangent of 42 degrees is the opposite 00:13:36.909 --> 00:13:50.935 side, x plus h, over the adjacent side, 300. So look at what we have. We have two equations with two unknowns. 00:13:50.937 --> 00:13:58.618 We're going to use some algebra to solve for our missing information. Ultimately, we need to find the height 00:13:58.620 --> 00:14:05.447 of the monument. Remember we called that h, so our goal is to find this h. We're going to use substitution. 00:14:05.449 --> 00:14:16.949 Take this first equation. Solve it for x. How do we do that? Multiply both sides by 300, so x is 300 times the 00:14:16.951 --> 00:14:27.466 tangent of 37.8 degrees. Before I do any substituting, I'm going to do something to this equation as well. 00:14:27.468 --> 00:14:43.200 I'd like to not have that fraction in my way, so I'm going to multiply both sides of this one by 300 as well. 00:14:43.202 --> 00:14:56.233 Ok, what can I do? Substitution; x is this expression. I'm just going to substitute this expression for the x, 00:14:56.235 --> 00:15:24.020 right here. So x is 300 times the tangent of 37.8 degrees. This should be a plus sign. This is x plus h, and to 00:15:24.022 --> 00:15:35.040 get h by itself, I need to subtract this expression from both sides. So h is going to be 300 tangent 42 degrees 00:15:35.042 --> 00:15:44.533 minus 300 times the tangent of 37.8 degrees. Now at this point, you're just using your calculator to 00:15:44.535 --> 00:15:51.301 approximate it. You can do that from here. You can simplify it first by factoring out the 300 if you choose to. 00:15:51.303 --> 00:16:13.575 Either way, calculator approximation. So I get 37.417. We were told to round to one decimal place again, so that's 00:16:13.577 --> 00:16:30.500 about 37.4 feet. Practice working through these examples until you've mastered the skills and concepts involved.