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Off on a Tangent

We’ve divided the side opposite the angle by the hypotenuse, and the side adjacent to the angle by the hypotenuse. What have we missed?

Well, we haven’t yet tried dividing the side opposite by the side adjacent.

This third combination is called the tangent, abbreviated as tan. Here’s a draggable triangle to experiment with.


[Applet appears here for Java-capable browsers]

Let’s see what kind of trail the tangent function leaves:


[Applet appears here for Java-capable browsers]

Yikes! The tangent is nowhere near as "mellow" as sine and cosine. Look at what happens as you approach 90 degrees. The cosine (the side adjacent) gets very small, so the result of the division starts getting very large.

At exactly 90 degrees, the cosine becomes zero, and you can't divide by zero. This accounts for the “spike” at 90 degrees (and also at 270 degrees).

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Surveying the Uses of Trigonometry >>