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Secant, cosecant and cotangent

Secant, cosecant and cotangent

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Explainer Video

Tutor: Toby

Summary

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)=1sin⁡(x)\boxed{\cosec(x)=\frac1{\sin(x)}}cosec(x)=sin(x)1​​​​

​

​sec⁡(x)=1cos⁡(x)\boxed{\sec(x)=\frac1{\cos(x)}}sec(x)=cos(x)1​​​​


cot⁡(x)=1tan⁡(x)\boxed{\cot(x)=\frac1{\tan(x)}}cot(x)=tan(x)1​​​​


Note: These trigonometric functions do not have a button on the calculator, so you have to use the sin⁡\sinsin, cos⁡\coscos and tan⁡\tantan functions.


Example 1

Use your calculator to find cot⁡(48∘)\cot(48^\circ)cot(48∘)​ to 3 s.f.3\text{ s.f.}3 s.f.


With your calculator in degrees mode, type in 1tan⁡(48)\frac1{\tan(48)}tan(48)1​:

cot⁡(48)=1tan⁡(48)=11.111=0.900‾ (to 3 s.f.)\cot(48)=\frac1{\tan(48)}=\frac1{1.111}=\underline{0.900}\text{ (to }3\text{ s.f.})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∘)\cosec(210^\circ)cosec(210∘).


First identify the trigonometric function that you will be able to use. Since cosec⁡(210)=1sin⁡(210)\cosec(210)=\frac1{\sin(210)}cosec(210)=sin(210)1​, you will find sin⁡(210∘)\sin(210^\circ)sin(210∘) next. To do so, use that 210=180+30210=180+30210=180+30. By using the sine graph you have

Maths; Trigonometric functions; KS5 Year 13; Secant, cosecant and cotangent

sin⁡(210∘)=−sin⁡(30∘)=−12\sin(210^\circ)=-\sin(30^\circ)=-\frac{1}2sin(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+(12)(−1)=−12\begin{aligned}\sin(210)&=\sin(180+30)\\&=\sin(180)\cos(30)+\sin(30)\cos(180)\\&=0+\left(\frac12\right)(-1)\\&={-\frac12}\end{aligned}sin(210)​=sin(180+30)=sin(180)cos(30)+sin(30)cos(180)=0+(21​)(−1)=−21​​​​


Hence you have that cosec⁡(210∘)=1−12=−2‾\cosec(210^\circ)=\dfrac1{-\frac12}=\underline{-2}cosec(210∘)=−21​1​=−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:

cos‾ec(x)=1s‾in(x)sec‾(x)=1c‾os(x)cot‾(x)=1t‾an(x)\begin{aligned}\text{co\underline{s}ec}(x)&=\frac1{\text{\underline{s}in}(x)}\\\text{se\underline{c}}(x)&=\frac1{\text{\underline{c}os}(x)}\\\text{co\underline{t}}(x)&=\frac1{\text{\underline{t}an}(x)}\\\end{aligned}cos​ec(x)sec​(x)cot​(x)​=s​in(x)1​=c​os(x)1​=t​an(x)1​​​​





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FAQs - Frequently Asked Questions

How do you remember the reciprocal trigonometric functions?

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.

What are the reciprocal trigonometric functions?

cosec(x), sec(x) and cot(x). Respectively, they are the reciprocals of sin(x), cos(x) and tan(x).

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