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In this lesson, you will cover what tangent, cotangent, secant, and cosecant are used for and how they are used. You will also learn to use the basic trigonometric ratios that were mentioned in an earlier section to help you solve for cotangent, secant, and cosecant.
There are six major trigonometric ratios that can help you to solve for lengths of sides in right triangles. The three that we have covered so far are as follows:
The tangent function was mentioned earlier when we first introduced trigonometric ratios. The tangent function can be written as:
This makes sense because, from the trigonometric ratios, we know that 
, and 
. Since 
, we can use substitution to find that:
which gives us:
And finally, after cancellation, we have:
.
Since “-toa,” the trigonometric ratio from soh cah toa, is 
, we have just proven that
. The cotangent is 
.
In terms of sine and cosine:
We mentioned earlier that there are six trigonometric ratios. The cotangent is another one of the trigonometric ratios, and can be written:
Tangent and cotangent functions are also referred to as ratio identities since 
and 
.
While sine and cosine can be considered the two most important trigonometric functions (after all, every trigonometric function can be written in terms of sine and cosine), it is still important to recognize the tangent and cotangent. The tangent, in particular, has some great uses that will be mentioned in a future lesson.
On the unit circle below, the tangent is located on segment 
.
Because you need to use an adjacent side, use triangle AFE. You know that, in triangle AFE, the side adjacent to 
is 1.
On the unit circle above, the cotangent is segment 
.
Because you need to use an opposite side, use triangle ADG. You know that, in triangle ADG, the side opposite of 
is 1.
The secant is equal to one over the cosine, and can be written
The trigonometric ratio for secant is
The cosecant is similar to the secant, and can be written as
The cosecant is the last important trigonometric ratio that you will need to be familiar with. It can be expressed as
The cotangent, secant, and cosecant are also known as the reciprocal identities because 
, and 
.
On the unit circle below, the secant is the segment 
.
Because you need to use an adjacent side, use triangle AEF. You know that, in triangle AEF, the side adjacent to 
is 1.
On the unit circle above, the cosecant is the segment 
.
Because you need to use an opposite side, use triangle ADG. You know that, in triangle ADG, the side opposite of
is 1.
You should now have a working understanding of all six trigonometric ratios.
For a quick refresher, the trigonometric ratios are:
| 1 |  |
4 |  |
| 2 |  |
5 |  |
| 3 |  |
6 |  |
The ratio identities are:
The reciprocal identities are:
The new trigonometric ratios are:
All six trigonometric functions can be derived in terms of sine and cosine.
