The angle of depression from an observer in an apartment...

The angle of depression from an observer in an apartment complex to a gargoyle on the building next door is $55^{\circ}$. From a point five stories below the original observer, the angle of inclination to the gargoyle is $20^{\circ}$. Find the distance from each observer to the gargoyle and the distance from the gargoyle to the apartment complex. Round your answers to the nearest foot. (Use the rule of thumb that one story of a building is 9 feet.)

Key Concepts

Trigonometric Ratios

Trigonometric ratios such as sine, cosine, and tangent are used to relate the angles of the right triangle to its side lengths. In many cases, the tangent ratio (opposite over adjacent) is especially useful when dealing with known vertical distances and unknown horizontal distances, while the sine and cosine ratios are helpful when the hypotenuse is required. These ratios provide the mathematical tools needed to compute the distances from the observers to the gargoyle as well as between the buildings.

Unit Conversion

Since the building stories are given in a non-standard unit (stories), converting these into feet is crucial for a consistent measurement system. By using the conversion factor (one story equals 9 feet), vertical distances can be accurately determined, which is necessary for applying trigonometric formulas to find the horizontal and slant distances.

Angle of Depression and Angle of Inclination

These angles are measured relative to an observer’s horizontal line of sight. The angle of depression is the downward angle from the horizontal, and by alternate interior angles in parallel lines, it equals the angle of elevation from the target point upward to the observer. The angle of inclination is essentially the angle of elevation from the observer looking up at an object, allowing us to model the situation using right triangles.

Right Triangle Geometry

The problem can be modeled using right triangles, where vertical distances (e.g., the height differences between floors) and horizontal distances (e.g., the distance between buildings) form the two perpendicular sides. The observer’s line of sight serves as the hypotenuse of these triangles. Understanding this geometry is essential for setting up the trigonometric relationships that help in finding unknown distances.

Transcript

00:01 So we have a building here and we have a building here and there's a gargoyle here.

00:06 And we know that this person's angle of depression down is supposed to be 55 degrees.

00:17 And then if we go down five stories and each story is supposed to be about nine feet, so if we go down about 45 feet, this angle of inclination, of incline, is supposed to be 20 degrees to the gargoyle.

00:36 And we want to find how far this is, how far this is away, and we also want to know this distance.

00:42 And i'll put that distance in after we do our analysis here.

00:46 So let me draw a little perpendicular horizontal line.

00:51 And we know alternate interior angles that this angle right here, i'll mark it in red, is supposed to be 55 degrees.

01:00 We also know that this little angle alternate interior that is green is supposed to be 20 degrees because these two will be congruent and these two will be congruent.

01:10 I just didn't draw it to make it look that way so this angle is supposed to be 75 degrees so we have a side angle pair and we can figure out these other angles because we know that this angle is going to be 35 degrees because these two have to be complementary and and this angle must be 70 degrees because these two angles must be complementary.

01:35 So we have all the angles, and again, if we add these two together, we get 105 degrees, plus the 75 degree angle does add up to 180 degrees.

01:47 So let's set up our law of signs, and we'll go x is to the sign of the angle opposite, and the sign of the angle opposite is the 70 degree angle.

01:57 And now i'm going to put the one that we know the pair.

02:01 We know 45 feet is to the sign of the angle opposite, which is that sign of 75 degrees.

02:10 And then we want to find y as well...

Please give Ace some feedback