From the top of a fence, a person sites a lion on the ground at an angle of depression of 24 degrees. If the man and the fence is 4.2 meters high , how far is the man from the lion ?

Answer 1

The first step should be to draw a diagram of the situation. You should end up with a right triangle with the lion and man at the ends of the hypotenuse, the fence as one side, and the ground between the fence and the lion as the other side. You know the angle of elevation (the angle where the lion is, which is the same as the angle of depression from the man to the lion) and the length of the opposite side (the fence). The distance from the man to the lion is the length of the hypotenuse, and the relationship between that value and the other two values you know is a simple trig function.

Answer 2

The tricky thing about angles of depression is that it's not what you might assume it is. If the person is sitting on something up high, think about their horizontal line of sight - a horizontal line going from their eyes out to the horizon. Then have them look down at the object (in this case, the lion). The angle between the horizontal line of sight and the line from the person to the lion is the angle of depression, in this case 24 degrees.

So if you were to draw a triangle from the person (top of fence, P) to the lion (L) to the bottom of the fence (F), the right angle is at angle F. Angle P in the triangle is going to be 90 - 24 = 66 degrees, because we're finding the angle inside the triangle, and 24 degrees was outside the triangle. The fence height (FP) is given, 4.2m. We are asked to find the hypotenuse, the line from the person to the lion (PL).

Since we know angle P, and the height FP is adjacent to angle P, and we need to find the hypotenuse, we'll use cosine, adjacent over hypotenuse.

Set up the algebraic equation: cos 66 = 4.2/x

Multiply both sides by x

x cos 66 = 4.2

Divide both sides by cos 66

x = 4.2/cos 66 = 4.2/0.41 = 10.2m