Question

Comment on the similarities and differences for the algebraic model (equation) of every trigonometric (sinusoidal or periodic) function.

Answer 1

Let's fist talk about the sine and cosine function. We know that both functions are periodic with period 2pi, that is:

`\begin{gathered} \sin (x+2\pi)=\sin x \\ \cos (x+2\pi)=\cos x \end{gathered}`

We also know that both functions are bounded in the intervale [-1,1], which means that:

`\begin{gathered} -1\leq\sin x\leq1 \\ -1\leq\cos x\leq1 \end{gathered}`

Both the sine and cosine functions have defined parity, the sine function is odd and the cosine function is even, that is:

`\begin{gathered} \sin (-x)=-\sin x \\ \cos (-x)=\cos z \end{gathered}`

Now, let's talk about the remaining trigonometric functions. They are defined as:

`\begin{gathered} \tan x=(\sin x)/(\cos x) \\ \cot x=(\cos x)/(\sin x) \\ \sec x=(1)/(\cos x) \\ \csc x=(1)/(\sin x) \end{gathered}`

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:

`\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}`

They also have defined parity. The secant is even and the remaining three functions are odd, that is:

`\begin{gathered} \tan (-x)=-\tan x \\ \cot (-x)=-\cot x \\ \sec (-x)=\sec x \\ \csc (-x)=-\csc x \end{gathered}`

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.