Question

find the range and domain of cosecant function​

Answer 1

Domain

Given w(θ) = (x,y)

$\large\rm$

$\large\rm{ we \ know \ that \ \csc \theta = \dfrac{1}{y}}$

but $\large\rm{ \dfrac{1}{y}}$ is undefined when y = 0.

csc(θ) is undefined for θ $\large\rm{ ....., -2 \pi , - \pi , 0 , \pi , 2 \pi}$

therefore domain of function is

$\large\rm{ dom(\csc) = \bigcup\limits_{ k \in \mathbb{Z}} ( k \pi, ( k+1) \pi )}$

Range

on the right semicircle, x ranges from 1 down to 0, so 1/x ranges from 1 upto ∞.

on left semicircle, x ranges from near 0 to -1, so 1/x ranges from -∞ upto -1.

Therefore range of the function is

$\large\rm{ ( - \infty , -1] \cup [ 1, \infty) }$

Answer 2

Answer:

The graph of the cosecant function looks like this: The domain of the function y=csc(x)=1sin(x) is all real numbers except the values where sin(x) is equal to 0 , that is, the values πn for all integers n . The range of the function is y≤−1 or y≥1 .