WEBVTT
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We're going to take a look at solving right triangles. To solve a right triangle means to find the measure
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of each of the acute angles and the length of each of the three sides. It's a good idea to organize your
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information from the beginning. There are three sides. We need to know the length of each. c is the
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hypotenuse, 10, which happens to be given this time. Then we have three angles. Angle C is the 90 degree
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angle opposite the hypotenuse. B is given to be 35 degrees, and A we'll need to find. Now finding angle A is
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the easiest thing we can do in this problem because we know that the sum of the three angles in a triangle
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is 180 degrees. So A is 180 minus the 35, minus the 90 degrees, and if we subtract this we should come up
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with 55 degrees. Ok, so what are we going to do next? We need at least one more side before we can
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think about the Pythagorean theorem or anything, so we're going to have to use trig. We'd like to use one
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of those six trig ratios that we've learned. We have some choices. Let's find b first. [Side] b is the side opposite
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35 degrees, so which ratio might I use? It kind of depends on what else I know. I'd like to know the
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opposite side. I do know the hypotenuse. So now we know which ratio to use, right? The sine of 35 degrees
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is the opposite side, b, over the hypotenuse. We're going to solve for b. Multiply both sides by 10.
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So we need our calculator. Make sure it's in degree mode, and we want to multiply 10 times the sine of
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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.
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We have one more piece of information to find, and that is a. We could use the Pythagorean theorem
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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
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degrees, and we know the hypotenuse is exactly 10. So we're going to go ahead and use the cosine of 35
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degrees to find a. So it's going to be a over 10. Multiply both sides by 10 again. Use your calculator to
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approximate that to one decimal place. This is going to be approximately 8.2 rounded to one decimal place.
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Let's try one more.
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In this right triangle, we're given two sides, and we'd like to solve that right triangle. So here we need to
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find the third side and the two acute angles. Let's start by organizing our information. So a is 5, b is 6,
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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
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we know that C is our right angle. We can find the hypotenuse, c, using the Pythagorean theorem, so let's
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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.
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We use our calculator to approximate that to one decimal place, and it's about 7.8. Now for the angles we'll
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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
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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
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tangent on our calculator to approximate A. Inverse tangent is usually that second function on your tangent
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button. This is about 39.8 degrees. You need to make sure your calculator is in degree mode for that one.
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Ok, the only thing left is angle B. It's one of the acute angles in the right triangle. We know that the sum
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of all those angles is 180 degrees, so that means the sum of these acute angles is 90 degrees.
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The easiest way to find B then is to take 90 minus the 39.8 degrees you just found for angle A,
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and we find out then that B is 50.2 degrees.
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Solving right triangles has many applications. Here's one. A hawk is sitting on top of a 34 foot pole spots
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a mouse on the ground at an angle of depression of 28 degrees. Find the line of sight distance between
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the hawk and the mouse. So clearly before we start we need to draw some sort of a picture. Here's our
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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
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pole, there's our hawk, and he's looking, and he finds a mouse. The mouse is down here. The question
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asks us to find the line of sight distance between the hawk and the mouse. So we want to know how far
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is it for the hawk to get to the mouse. Ok, well we have one other piece of information we need to place
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on our picture. The angle of depression from the hawk looking down to the ground is 28 degrees. An angle
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of depression is relative to the horizontal, and it's depression. It's going down. So this is our angle of
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28 degrees that we're given. This dotted line, the diagonal one here, that is the line of sight distance between
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the hawk and the mouse, and that is what we want to find. So let's call it x. Where's the right triangle?
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Well we have to assume that the pole is perpendicular to the ground, so there we go. Ok, well there's one
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other problem. Notice that the 28 degrees is not part of the right triangle that I would like to solve, so we're
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going to do a little geometry. If this is a right angle, then so is this one, so that means that the 28 degrees
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and alpha have to have a sum of 90 degrees. So the first thing I'm going to do is subtract 90 minus 28,
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which is 62 degrees, so that's alpha. I can just stop there because alpha is in the right triangle. It's one
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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
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right triangle, and I know this leg. This leg is adjacent to angle alpha, and that should trigger cosine.
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Cosine of this angle is the adjacent side over the hypotenuse. So let's write this down. The cosine of 62 degrees
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is 34 over x. We know how to solve this, right? Multiply both sides by x. Divide both sides by cosine 62
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degrees, and we take out our calculator and approximate this to one decimal place as requested.
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So about 72.4 feet is the line of sight distance from the hawk to the mouse.
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This next example is a little bit more complicated. We have to use two right triangles to find a missing
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piece of information. A monument is spotted at the edge of a cliff. At a distance of 300 feet from the base
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of the cliff, the angle of elevation to the bottom of the monument is 37.8 degrees, and the angle of elevation
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to the top of the monument is 42 degrees. Find the height of the monument rounded to one decimal place.
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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
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a monument of some person on the top of the cliff. Now imagine that you're standing over here, 300 feet
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away from the cliff, and you look up to the bottom of the monument. We were told the angle of elevation to
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the bottom of the monument is 37.8 degrees. The angle of elevation to the top of the monument, from that
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same place, is 42 degrees. We're going to assume the cliff is perpendicular to the ground, so we have a right
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triangle here. We were told we were standing 300 feet away, so let's label 300 feet. This is really all we know,
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but we do have some unknowns that will be helpful to us. We're asked to figure out the height of the monument,
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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.
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We have two right triangles. We know one acute angle. Look at the 300 feet. That is the adjacent side to the
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37.8 degrees, and it's also the adjacent side, in the bigger triangle, to the 42 degrees. Let's concentrate on one
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of those triangles to begin with. Let's concentrate on the smaller one. What side is opposite 37.8 degrees?
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That's our x that we don't know, and the adjacent side is 300. So we're going to use tangent, opposite
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over adjacent. Tangent of 37.8 degrees is x over 300. Now let's look at the bigger triangle. This one has an
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angle of elevation of 42 degrees. The adjacent side is still 300, and the opposite side is all of this, the cliff
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plus the monument, or x plus h. We can write another equation. Tangent of 42 degrees is the opposite
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side, x plus h, over the adjacent side, 300. So look at what we have. We have two equations with two unknowns.
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We're going to use some algebra to solve for our missing information. Ultimately, we need to find the height
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of the monument. Remember we called that h, so our goal is to find this h. We're going to use substitution.
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Take this first equation. Solve it for x. How do we do that? Multiply both sides by 300, so x is 300 times the
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tangent of 37.8 degrees. Before I do any substituting, I'm going to do something to this equation as well.
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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.
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Ok, what can I do? Substitution; x is this expression. I'm just going to substitute this expression for the x,
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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
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get h by itself, I need to subtract this expression from both sides. So h is going to be 300 tangent 42 degrees
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minus 300 times the tangent of 37.8 degrees. Now at this point, you're just using your calculator to
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approximate it. You can do that from here. You can simplify it first by factoring out the 300 if you choose to.
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Either way, calculator approximation. So I get 37.417. We were told to round to one decimal place again, so that's
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about 37.4 feet. Practice working through these examples until you've mastered the skills and concepts involved.