Chapter 4.6, Problem 1E

To determine

To fill: in the blanks.

Answer to Problem 1E

The tangent, cotangent and cosecant functions are odd, so the graphs of these functions have symmetry with respect to the origin”.

Explanation of Solution

Given information:

The tangent, cotangent, and cosecant functions are ----------, so the graphs of these functions have symmetry with respect to the --------------.

Calculation:

For the given three functions; tangent, cotangent and cosecant

  $\begin{array}{l}\tan \left(-x\right)=-\tan x\\ \cot \left(-x\right)=-\cot x\\ \sec \left(-x\right)=-\sec x\end{array}$

Such a function is known as odd function. So, the tangent, cotangent and cosecant functions are odd.

Now, when we plot the graphs for the odd functions; for example,

  $y=\tan x$

  

The function has symmetry with respect to the origin. Same is the case with cotangent and cosecant. So, the graphs of these functions have symmetry with respect to the origin.

Therefore,

“The tangent, cotangent and cosecant functions are odd, so the graphs of these functions have symmetry with respect to the origin”.