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The thing that looks like the hypotenuse of a right triangle is called a vector . It has a length and it has a direction. Typically, the direction is stated as the angle between the vector and the horizontal axis. Thus, thedirection is analogous to the angle at the origin in the triangle on your graph board.

Horizontal and vertical components

For reasons that I won't explain until we get to that module, you will often need to compute the horizontal and vertical components of the vector.The horizontal component is essentially the adjacent side of our current right triangle. Thus, the value of the horizontal component can be computed using thesecond equation in Figure 10 .

The vertical component is essentially the opposite side of our current right triangle, and its value can be computed using the first equation in Figure 10 .

The tangent and arctangent of an angle

Once again, although the tangent of an angle is based on very specific geometric considerations involving circles (see (External Link) ), for our purposes, the tangent of an angle is simply a ratio between the lengths of two different sides of a righttriangle.

A ratio of two sides

For our purposes, we will say that the tangent of an angle is equal to the ratio of the opposite side and the adjacent side. Therefore, in the case of the3-4-5 triangle that you have on your graph board, the tangent of the angle at the origin is equal to 4/3 or 1.333.

Not limited to 1.0

Note that the absolute value for the sine and the cosine of an angle is limited to a maximum value of 1.0. However, the tangent of an angle is not solimited. In fact, the tangent of 45 degrees is 1.0 and the tangent of 90 degrees is infinity. This results from the length of the adjacent side, which is thedenominator in the ratio, going to zero at 90 degrees.

Dividing by zero in a script is usually not a good thing. This is a pitfall that you must watch out for when working with tangents. I will provide code later on that shows you how deal with this issue.

Computing the tangent

If we know the lengths of the opposite side and the adjacent side, we can compute the tangent and use it for other purposes later without having to knowthe value of the angle.

Conversely, if we know the value of the angle but don't know the lengths of the adjacent side and/or the opposite side, we can obtain the tangent valueusing a scientific calculator or lookup table and use it for other purposes later.

The tangent of an angle -- sample computation

Enter the following into the Google search box:

tan(53.13010235415598 degrees)

The following will appear immediately below the search box:

tan(53.13010235415598 degrees) = 1.33333333

This agrees with the ratio that we computed earlier .

The arctangent (inverse tangent) of an angle

The arctangent of an angle is the value of the angle having a given tangent value. (For example, as mentioned above, the arctangent of infinity is 90degrees and the arctangent of 1.0 is 45 degrees.) In other words, if you know the value of the tangent of an unknown angle, you can use a scientific calculator or lookup table to find the value of theangle.

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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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