SOLUTION: The angle of elevation from the top of a house to a jet flying 2 miles above the house is x radians. If d represents the horizontal distance, in miles, of the jet from the house, e

Algebra ->  Trigonometry-basics -> SOLUTION: The angle of elevation from the top of a house to a jet flying 2 miles above the house is x radians. If d represents the horizontal distance, in miles, of the jet from the house, e      Log On


   

Question 352684: The angle of elevation from the top of a house to a jet flying 2 miles above the house is x radians. If d represents the horizontal distance, in miles, of the jet from the house, express d in terms of a trigonometric function of x. Then graph the function for 0 < x < Pi.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
It will probably help to have a drawing/diagram of the problem. Start with a point which we will name A. From A draw a horizontal line segment (the length of the segment is not important). At the end of this segment we will name the point B. From B draw a vertical line segment (again, the length of the segment is not important). At the end of this segment we will name the point C. Then draw a segment connecting points A and C. Write an "x" inside the angle formed at point A, write a "d" below segment AB and halfway between A and B and write a "2" next to segment BC and halfway between B and C.

We should now have a right triangle with the right angle at B (because horizontal and vertical are perpendicular). At A we have an angle of x radians. Segment BC has a length of 2 because the jet is 2 miles up. The distance we are interested in is "d", the horizontal distance.

To come up with a trigonometric equation for "d" we have to know the ratios represented by the various trig functions. Then we have to select the right one for this problem.

Looking at the diagram we should be able to tell that from the point of view of the ange at A, segment BC is the "opposite" side and that segment AB is the adjacent side. So we need a ratio involving opposite and adjacent sides. If we know the trig functions and their ratios we know that both tan (tangent) and cot (cotangent) are ratios of opposite and adjacent.

For reasons you will see shortly, I am going to pick the function which has the adjacent side in the numerator of the ratio. This would be cot. Writing an equation which connects tan to these particular opposite and adjacent sides we get:
cot%28x%29+=+d%2F2
To solve for d, all we have to do is multiply by 2! (This is why I chose cot. With d in the numerator it is much easier to solve for.)
2%2Acot%28x%29+=+d
The full graph of this looks like:
graph%28400%2C+400%2C+-4%2C+4%2C+-4%2C+4%2C+2%2Ftan%28x%29%29
(I'll leave it up to you to truncate this graph to the interval of (0, pi).