Angles of Elevation / Inclination and
     Angles of Depression / Declination

Angles of elevation or inclination are angles above the horizontal, like looking up from ground level toward the top of a flagpole. Angles of depression or declination are angles below the horizontal, like looking down from your window to the base of the building in the next lot. Whenever you have one of these angles, you should immediately start picturing how a right triangle will fit into the description.

    The scale is not important; I'm not bothering to get the angle "right". I'm using the drawing as a way to keep track of information; the particular size is irrelevant.

    What is relevant is that I have "opposite" and "adjacent" and an angle measure. This means I can create and solve an equation:

      h/1056 = tan(2.4°)
      h = 1056×tan(2.4°) = 44.25951345...


right triangle, base angle 2.4*, base / adjacent 1056, height / opposite h

    First, I draw my triangle:

    The horizontal line across the top is the line from which the angle of depression is measured. But by nature of parallel lines, the same angle is in the bottom triangle. I can "see" the trig ratios more easily in the bottom triangle, and the height is a bit more obvious. So I'll use this part of the drawing.


right triangle with base angle 15*, hypotenuse 325, opposite / height h

    I have "opposite", hypotenuse, and an angle, so I'll use the sine ratio to find the height.

      h/325 = sin(15°)
      h = 325×sin(15°) = 84.11618966...

    The bluff is about 84 feet above the lake.


  • A lighthouse stands on a hill 100 m above sea level. If ∠ACD measures 60° and ∠BCD is 30°, find the height of the lighthouse.


drawing of hill, tower, and angles; A at top of tower, B at base, D at base of hill, C common vertex

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