Secant, cosecant and cotangent
In a nutshell
These three trigonometric functions are called the reciprocal trigonometric functions because of their reciprocal relationships with the functions sine, cosine and tangent, respectively.
Definitions
cosec(x)=sin(x)1
sec(x)=cos(x)1
cot(x)=tan(x)1
Note: These trigonometric functions do not have a button on the calculator, so you have to use the sin, cos and tan functions.
Example 1
Use your calculator to find cot(48∘) to 3 s.f.
With your calculator in degrees mode, type in tan(48)1:
cot(48)=tan(48)1=1.1111=0.900 (to 3 s.f.)
Example 2
Without using a calculator, find the exact value of cosec(210∘).
First identify the trigonometric function that you will be able to use. Since cosec(210)=sin(210)1, you will find sin(210∘) next. To do so, use that 210=180+30. By using the sine graph you have
sin(210∘)=−sin(30∘)=−21
Alternatively, you could have used the double angle formula:
sin(210)=sin(180+30)=sin(180)cos(30)+sin(30)cos(180)=0+(21)(−1)=−21
Hence you have that cosec(210∘)=−211=−2.
Tool to remember
One way to remember which reciprocal trigonometric function relates to which trigonometric function is to look at the third letter of the reciprocal function. It corresponds to the first letter of the related trigonometric function:
cosec(x)sec(x)cot(x)=sin(x)1=cos(x)1=tan(x)1