Determining an Angle from Its Tangent

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Determining an Angle from Its Tangent

In the figure above, you are looking at an object and there is an angle φ between the lines to the top and bottom of the object. When the objects are very distant, a very close approximation can be more easily made by using the measurements drawn in red. (It will be left to high school students in Euclidean geometry to show that this approximation is indeed very close when the distance between the object and eye is very large relative to the size of the object.)

Using the Approximate Method
(But you might want to read the "GREEN" section, below, first!)

The tangent of an angle is defined at the height of the object divided by the distance. Your experimental data has provided you with both the height and distance. Now what do you do with those to determine the angle? Here are several ways to do it:

  1. Using your pocket calculator
    1. Testing your calculator
      1. Using the buttons on your calculator, input: TAN 1 ENTER
      2. If the answer that appears is 1.557..., then your calculator uses radians and you should use the PURPLE section below.
      3. If the answer that appears is 0.017..., then your calculator uses degrees and you should use the RED section below.
    2. If your calculator uses degrees
      1. If necessary, calculate your average height and your average distance.
      2. Input: 2nd TAN ( height ÷ distance ) ENTER
      3. OR you might want to use the "GREEN" method below.
    3. If your calculator uses radians
      1. If necessary, calculate your average height and your average distance.
      2. Input: 57.295 x ( 2nd TAN ( height ÷ distance ) ) ENTER
      3. OR you might want to use the "GREEN" method below.

  2. Using published tables of angles and their tangents
    1. If necessary, calculate your average height and your average distance.
    2. Calculate: height ÷ distance
    3. Go to your table of tangents, and look for that number in the tangent column. You will probably only see a number that is very close to your number. Using that number's line and go across to the angle column, and there's your approximate angle. BUT you may need to do the next step---
    4. If you are dealing with a small object at a great distance, the above will not be usable since you will be reading all your values between 0 degrees and the next value.
      • A commonly used approximation is that as the angle approaches 0.0000°, the angle approaches the value of height/distance. (How could things be simpler!)

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