Angles of Inclination / Declination

If there were a plank between where you're standing and that chimney's top, walking that plank would involve hiking up an incline. You must elevate your eyes (above the horizontal) to focus on that chimney. The angle above the horizontal that corresponds to your line of sight above the street between you and that house's chimney would be the angle of inclination (also called the angle of elevation).

What is an angle of depression / declination?

An angle of declination (also called the angle of depression) is an angle that goes below the horizontal from whatever is the vantage point. For instance, suppose you are standing on the balcony of your third-floor apartment and are wondering how far down the sidewalk is. The angle between the horizontal (being you looking across the parking lot to your friend on her third-floor deck in the next building over) and the sidewalk would be the angle of depression (also called the angle of declination).

What is the difference between angles of inclination/elevation and angles of declination/depression?

Angles of elevation/inclination are angles above the horizontal; angles of depression/declination are angles below the horizontal.

Whenever you have one of these angles, you should immediately start picturing how a right triangle will fit into the description.

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• Driving along a straight flat stretch of Arizona highway, you spot a particularly tall saguaro ("suh-WAHR-oh") cactus right next to a mile marker. Watching your odometer, you pull over exactly two-tenths of a mile down the road. Retrieving your son's theodolite from the trunk, you measure the angle of elevation from your position to the top of the saguaro as 2.4°. Accurate to the nearest foot, how tall is the cactus?

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• You were flying a kite from the top of a bluff, but you managed somehow to dump your kite into the lake below. You know that you've given out 325 feet of string. A surveyor on his lunch break wanders over and tells you that the angle of declination from your position to the kite is 15°. How high is the bluff where you and the surveyor are standing? (Round your answer to the nearest whole number.)

• A lighthouse stands on a hill 100 m above sea level, as shown in the graphic below. If ∠ACD measures 60° and ∠BCD is 30°, find the height of the lighthouse, to the nearest whole meter.

What are the steps for angles of inclincation/declination?

To sum up, the method for solving inclination/declination exercises is this:

1. By definition, an angle of inclination or declination is measured with respect to some horizontal line. Draw that line, and the angle with respect to it.
2. By definition, an angle of inclination or declination is relating that angle to the height (or depth) that an object is above (or below) the horizontal. Draw that height line. You should now have a right angle.
3. The line-of-sight will be the hypotenuse of the right triangle. Draw this line.
4. Solve the right triangle for the required information.