Welcome to the first of two videos
on the applications of the law of sines.
Let's go ahead and take a look at a problem.
Standing on one bank of the canyon,
a surveyor notices a tree at a bearings of 115 degrees.
Then the surveyor walks 300 meters.
The bearings of the tree is then 85 degrees.
What is the distance across the canyon?
Let's go ahead and make a sketch.
Let's say this is our canyon,
and this is the first position of the surveyor,
and this is the second position of the surveyor.
And on the other side of the canyon,
there's a tree on the edge of the canyon.
Remember, bearings are measured
from due north.
So if this is due north, the bearings
to this tree would be 115 degrees.
So this angle here is 115 degrees.
Then the surveyor walks 300 meters,
so the distance from this blue point
to this blue point is 300 meters.
And now the tree is at a bearings of 85 degrees,
so measuring from due north,
this angle here would be 85 degrees.
And the question wants us to determine
the distance across the canyon,
which would be...
This distance here, let's call it d.
Let's go ahead and take a look at our sketch
and see if we can fill in any additional information.
We know that this angle here of 115 degrees
and this angle here, inside this red triangle
would be supplementary.
So 180 minus 115 would give us 65 degrees for this angle.
Next, in the red triangle,
we know the sum of these three angles
must equal 180 degrees.
These two have a sum of 150 degrees,
so this angle here would be 30 degrees.
Now, we can't determine d in one step.
We're going to have to determine
either this distance here,
we'll call it x, or this distance here,
we'll call it y.
Once we find either x or y, we can use
a property of right triangles to determine d.
Let's go ahead and see if we can find
the length of this side here, which we're calling x,
using the law of sines.
Since we know this angle and the opposite side,
let's start with this ratio.
The sine of 30 degrees
divided by 300 meters
must equal the sine of 85 degrees
divided by x.
Now that we have a proportion,
we can cross-multiply.
x times sine 30 degrees
must equal 300 times sine of 85 degrees.
Now we can solve for x by dividing
by sine 30 degrees.
And now we'll go to the calculator.
300 sine 85 degrees divided by sine 30 degrees,
which is 1/2.
So x is approximately 598 meters.
Let's go ahead and label that.
Now, we should be able to use this right triangle
to determine the length of d,
or the distance across the canyon.
So write an equation that involves this angle,
this side, and this side.
And since this side is opposite this angle,
and this is the hypotenuse of that right triangle,
we can say the sine of 65 degrees
must equal the opposite side divided by the hypotenuse,
or d divided by 598.
If we put this over one, we have a proportion.
Let's go ahead and cross-multiply.
d times one would be d.
And 598 times sine 65 degrees would be...
Approximately 542 meters.
And that was the distance we were looking for,
the distance across the canyon.
Okay, we'll take a look at another problem
in the next video, but I hope you found this video helpful.
Thank you.