Elementary trigonometric functions

Elementary trigonometric functions

After having introduced the first three trigonometric functions of a given angle \beta, now we are going to introduce their reciprocals.

Secant:

This is the reciprocal of the \cos \beta noted sec \beta.

Let’s look at the following graph.

\triangle ADB' \sim \triangle ABE

\frac{AE}{AB'}=\frac{AB}{AD}

AB'=1

AB=1

AD=\cos \beta

\frac{AE}{1}=\frac{1}{\cos \beta}

AE=\sec \beta

Finally the secant is:

\sec \beta=\frac{1}{\cos \beta}

Cosecant:

This is the reciprocal of the \sin \beta noted csc \beta.

Let’s look at the following graph.

\triangle ADB' \sim \triangle FCA by AA

\frac{AF}{AB'}=\frac{CA}{DB'}

DB'=1

CA=1

DB'=\sin \beta

\frac{AF}{1}=\frac{1}{\sin \beta}

AF=\csc \beta

Finally the cosecant is:

\csc \beta=\frac{1}{\sin \beta}

Cotangent:

This is the reciprocal of the \tan \beta noted cot \beta.

Let’s look at the following graph.

\triangle ABE \sim \triangle FCA by AA

\frac{FC}{AB}=\frac{CA}{BE}

AB=1

CA=1

BE=\tan \beta

FC=\cot \beta

\frac{FC}{1}=\frac{1}{BE}

\frac{\cot \beta}{1}=\frac{1}{\tan \beta}

Finally the cotangent is:

\cot \beta=\frac{1}{\tan \beta}

We can also see that:

\triangle ADB' \sim \triangle FCA by AA

\frac{FC}{AD}=\frac{CA}{DB'}

CA=1

DB'=\sin \beta

AD=\cos \beta

FC=\cot \beta

We get:

\frac{\cot \beta}{\cos \beta}=\frac{1}{\sin \beta}

Finally:

\cot \beta=\frac{\cos \beta}{\sin \beta}

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