INSTRUCTOR: A pilot is flying over a straight highway. He determines the angles of depression to two towers 3.7 miles apart to be 67 degrees and 59 degrees. So the situation can be modeled by this diagram here. We're actually going to answer two questions based upon this diagram. The first one is to determine the length of segment AB or the distance from point A to point B. 

Let's start by determining all the information we can about this triangle. Since this highway is straight, and we'll assume the plane is flying straight, we have two parallel segments. And if we have parallel lines cut by a transversal, then alternate interior angles are equal or congruent. So this is 67 degrees, then this angle here would also be 67 degrees. And if this is 59 degrees, then this angle here is also 59 degrees. 

And then, looking at this last angle here, notice that these three angles form a straight angle. Therefore, their sum must be 180 degrees. So the measure of this angle here would be 180 minus 67 minus 59, which would be 54 degrees. So now, we have the measure of all the angles in this triangle, and our goal is to determine the length of side AB, which we'll label x. 

Looking at the information we have in the triangle now, notice we have the measure of this angle and the length of the opposite side. And this is an indication that we can solve for x using the law of sines. So now, we can set up a proportion with one unknown. We'll have the sine of 59 degrees divided by x must equal the sine of 54 degrees divided by 3.7 miles. 

And now, we can cross multiply and solve for x. So we'll have x sine 54 degrees must equal 3.7 times sine 59 degrees. And now, we can divide both sides by sine 54 degrees to determine x. So we'll have x is approximately equal to this quotient. 

So let's go to our calculator. Let's make sure that we're in degree mode. And we are. So our numerator is going to be 3.7 sine 59 degrees. We'll divide this by sine 54 degrees. So x is approximately 3.9. And that will be miles. 

Let's go ahead and label this. Now let's take a look at the second part of this question. Now we want to determine the elevation of the plane, which would be the length of this vertical segment here. Let's call it h. 

So when we sketch this height, notice we created two right triangles from this larger triangle. We can now use this small right triangle here to determine the height or altitude of this plane by using a trig equation. If we use the angle here that measures 67 degrees, h would be the opposite side, and this side here would be the hypotenuse. 

So we can write the trig equation sine 67 degrees must equal h divided by 3.9. And now, we can just multiply both sides of the equation by 3.9 to solve for h. That simplifies out. So h is equal to this product here, so we'll get a decimal approximation of 3.9 times the sine of 67 degrees. So the altitude of the height of that plane is approximately 3.6 miles.