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The tangent is the ratio of vertical height to horizontal distance in any context that can be represented with a right triangle. The tangent ratio provides useful information about steepness and can help us determine the measure of  , which is often referred to as the angle of elevation. Tables are available that show the relationship between the size of an angle and its tangent. Today, however, most people use the tangent key on a scientific calculator to obtain this information.
Experiment with your calculator, using the data in the table below for verification, to find the following:
• | The corresponding tangent, given an angle |
• | The corresponding angle, given a tangent. Note 8 |
We can use right-angle trigonometry to solve problems like those in Part A.
To determine the distance (b) of a tree (point C) across a river, first locate point B directly across the river (where the sighting line is perpendicular to the bank). Next we locate point A at some distance c on the same side of the river as B, and we physically measure the distance between the two (for example, 30 m). Standing at A, we sight C and measure (using a transit), which we determine to be 52 degrees:
Finally, we set up the tangent ratio to determine the distance between B and C:
tan 52° = b/c
Using this information, let's calculate the distance b.
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