Reciprocal Trigonometric Functions

Here is a brief overview of reciprocal trigonometric functions. It is assumed that the reader is familiar with \sin(\theta), \cos(\theta) and \tan(\theta). We work within the right angle triangles framework.


Reciprocal Trigonometric Functions

Reciprocal trigonometric functions may not be as common as the typical \sin(\theta), \cos(\theta) and \tan(\theta) but knowing them can be helpful.

The secant of an angle is one divided by the cosine of the angle. It is expressed as:

    \[\sec(\theta) = \dfrac{1}{\cos(\theta)}\]

Recall that the cosine of an angle is the side length adjacent to the angle divided by the length of the hypotenuse. The secant of an angle would then be length of the hypotenuse divided by the side length adjacent to the angle.


The cosecant of an angle is one divided by the cosine of the angle. It is expressed as:

    \[\csc(\theta) = \dfrac{1}{\sin(\theta)}\]

Remember that the sine of an angle is the side length opposite to the angle divided by the hypotenuse length of the right angled triangle. The cosecant of an angle would then be the hypotenuse length of the right angled triangle divided by the side length opposite to the angle.


The reciprocal of the tangent of an angle is the cotangent function. It is expressed as:

    \[\cot(\theta) = \dfrac{1}{\tan(\theta)}\]

Since the tangent of an angle is the side length opposite to the angle divided by the side length adjacent to the angle, the cotangent of an angle would be the reciprocal of the tangent ratio. (That is, the side length adjacent to the angle divided by the side length opposite to the angle.)

Alternatively, we can express the cotangent of an angle as:

    \[\cot(\theta) = \dfrac{\cos(\theta)}{\sin(\theta)}\]

using the fact that \tan{(\theta)} = \dfrac{\sin(\theta)}{\cos(\theta)}.


Here is an image which summarizes the sine, cosine, tangent functions and their corresponding reciprocal functions.

 

Source: http://1.bp.blogspot.com/-C05TaepcYb0/T1yh8m3cPTI/AAAAAAAAACU/20x2l-hfSvQ/s1600/trig.jpg

 


Examples

There are times when algebra and knowledge of how trigonometric functions work together can be useful in simplifying expressions.


Example One

    \[\csc(\theta) \tan(\theta) = \dfrac{1}{\sin(\theta)} \dfrac{\sin(\theta)}{\cos(\theta)} = \dfrac{1}{\cos(\theta)} = \sec{(\theta)}\]


Example Two

    \[\sec(\theta) \tan(\theta) = \dfrac{1}{\cos(\theta)} \dfrac{\sin(\theta)}{\cos(\theta)} = \dfrac{\sin(\theta)}{\cos^{2}(\theta)}\]


Example Three

    \[\sec(\theta) \cot(\theta) = \dfrac{1}{\cos(\theta)} \dfrac{\sin(\theta)}{\cos(\theta)} = \dfrac{\sin(\theta)}{\cos^{2}(\theta)}\]


Example Four

Source: http://quicklatex.com/cache3/bf/ql_ad0c837e837b2b859baebd1d1d0dfbbf_l3.png

 

In this example, we found a common denominator and simplified accordingly.


Example Five

You can also express \sin(\theta), \cos(\theta), and \tan(\theta) in terms of \csc(\theta), \sec(\theta) and \cot(\theta) respectively.

    \[\sin(\theta) = \dfrac{1}{\csc(\theta)}\]

    \[\cos(\theta) = \dfrac{1}{\sec(\theta)}\]

    \[\tan(\theta) = \dfrac{1}{\cot(\theta)}\]

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