Respuesta :
Let's fist talk about the sine and cosine function. We know that both functions are periodic with period 2pi, that is:
[tex]\begin{gathered} \sin (x+2\pi)=\sin x \\ \cos (x+2\pi)=\cos x \end{gathered}[/tex]We also know that both functions are bounded in the intervale [-1,1], which means that:
[tex]\begin{gathered} -1\leq\sin x\leq1 \\ -1\leq\cos x\leq1 \end{gathered}[/tex]Both the sine and cosine functions have defined parity, the sine function is odd and the cosine function is even, that is:
[tex]\begin{gathered} \sin (-x)=-\sin x \\ \cos (-x)=\cos z \end{gathered}[/tex]Now, let's talk about the remaining trigonometric functions. They are defined as:
[tex]\begin{gathered} \tan x=\frac{\sin x}{\cos x} \\ \cot x=\frac{\cos x}{\sin x} \\ \sec x=\frac{1}{\cos x} \\ \csc x=\frac{1}{\sin x} \end{gathered}[/tex]Since this four functions are defined from the sine and cosine function they will inherite some properties from this functions. The tangent, cotangent, secant and cosecant are all periodic functions; the first two have a period of pi and the reamining two have a period of 2pi, then we have:
[tex]\begin{gathered} \tan (x+\pi)=\tan x \\ \cot (x+\pi)=\cot x \\ \sec (x+2\pi)=\sec x \\ \csc (x+2\pi)=\csc x \end{gathered}[/tex]They also have defined parity. The secant is even and the remaining three functions are odd, that is:
[tex]\begin{gathered} \tan (-x)=-\tan x \\ \cot (-x)=-\cot x \\ \sec (-x)=\sec x \\ \csc (-x)=-\csc x \end{gathered}[/tex]One property that this functions don't inherite is that they are not bounded functions.
Let's sum up what we learn so far.
Similarities:
All the trigonometric functions are periodic and have defined parity.
Differences:
The sine and cosine functions are bounded while the reamining functions are unbounded.
