BIRD WATCHING Two observers are 200 feet apart, in line with a tree containing a bird's nest. Th

BIRD WATCHING Two observers are 200 feet apart, in line with...

Key Concepts

Solving Systems of Equations

In problems with multiple observations, a system of equations is frequently set up by applying the tangent function from different observation points. Solving these equations simultaneously enables the determination of unknown distances and heights.

Tangent Function

The tangent function, defined as the ratio of the opposite side to the adjacent side in a right-angled triangle, is particularly useful when an angle of elevation and the distance on the horizontal ground are involved. It directly relates the height of an object to the distance from the object.

Angle of Elevation

The angle of elevation is the angle formed between the horizontal line of sight and the line from the observer's eye to an object above the observer. It is a crucial concept in trigonometry and is used to determine heights or distances when combined with trigonometric functions.

Right Triangle Trigonometry

Right triangle trigonometry involves the relationships between the angles and sides of a right-angled triangle. In problems where an object’s height is observed from a certain distance, a right triangle is formed which allows the application of trigonometric ratios to find unknown lengths.

Transcript

00:01 So we have two observers, and they're bird watching, they're birding.

00:06 So we know that our two observers, which have put as dots here, are 200 feet apart, and they're both looking at a tree, at a nest in a tree, but we don't know the height of that tree.

00:21 So we'll just call that y right now.

00:23 What we do know is that the closer observer is looking up with an angle of elevation of 60 degrees, and the further away observer is looking up with an angle of elevation of 30 degrees.

00:36 And our goal is to find the distance from the observers to the base of the tree.

00:45 So what we want to do is use some right angle trigonometry.

00:48 We see that we have two right triangles here.

00:50 The first is made with the closer observer, the nest, and the base of the tree.

00:57 The other is made with the further away observer, the nest, and the nest, the base of the tree, right? so it seems like we don't know all that much information.

01:09 We only know that these two observers are 200 feet away and we know their angles, but we can use this information to determine the distance of each observer from the tree.

01:22 And to label that distance, that's going to be for the closer observer x1, this distance, and for the further away observer, we can call it x2.

01:32 But that's just going to equal x1 plus 200, right? because the further away observer is 200 feet away from the closer observer.

01:41 So we can use all this information to find two equations.

01:46 And with two equations, which will have variables y and x1, we can solve for x1.

01:56 And in doing so, we can also solve for x2.

01:58 So let's set those equations up.

02:00 The first equation, i'm going to redraw my green.

02:04 Triangle here the first equation is going to have to do with this green triangle.

02:09 Remember our height is y here our adjacent side is x1 and this angle of elevation is 60 degrees like this and this is all the information that we know about this triangle.

02:24 So it looks like we're going to have to use a trigonometric function that relates the opposite and adjacent side and that trigonometric function is tangent so we know that tangent of 60 degrees is equal to y over x1.

02:40 And that's as specific as we can get yet.

02:42 We don't know anything about why, so we can't solve for x1.

02:47 We could rearrange this equation a little bit to make it a little bit more useful to us in the future.

02:52 We know that, well, i guess we'll get to that a little bit later.

02:56 Right now, i'll examine the second triangle and make another equation here.

03:03 Our second triangle is similar.

03:06 It's not similar in a geometric sense, but we know some of the same information about it.

03:11 We know that the height is y and that this angle is 30 degrees.

03:16 This adjacent side is x2, but we said that that was equal to x1 plus 200.

03:23 So when we set up the same type of equation, we're going to use tangent to correlate x1 and our adjacent side.

03:33 Are opposite and adjacent sides.

03:36 So we have tangent this time of 30 degrees equals y over x1 plus 200.

03:46 And now we have two equations.

03:48 And we want to solve for x1.

03:52 So the way that we might do this, we notice that each one of these has a y.

03:57 So if we solve each equation for y, then we can set those equal to each other and use that information to solve for x.

04:05 X1.

04:06 So i'm going to show what i mean.

04:09 If tangent of 60 degrees equals y over x1, that means i can multiply both sides by x1 because i know x1 is not equal to zero.

04:19 And i get that x1 times tangent of 60 degrees equals y.

04:26 And i can do the same thing in this equation, the second equation, right? i can multiply both sides by x1 plus 200.

04:34 And i get that x1 plus 200 times tangent of 30 degrees also equals y.

04:46 So i know y is the same in both triangles, so i can set these equal to one another.

04:53 Right, so let's do that in another line here.

04:56 So what i have is that x1 times the tangent of 60 degrees equals x1 plus 200 times the tangent of 30 degrees...

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