Question

1. What is the domain of validity for csc theta=1/sin theta? A. All real numbers B. All real numbers except odd multiples of pi/2 C. All real numbers except even multiples of pi/2 D. All real numbers except multiples of pi

Answer 1

`\csc\Theta=(1)/(\sin\Theta)`

Since the function
`\csc\Theta` is literally defined as the reciprocal of
`\sin\Theta`, any value of
`\Theta` will satisfy this equation.

A. All real numbers

`\{\Theta|\Theta\in\mathbb{R}\}`

Answer 2

Option D. All real numbers except multiples of pi

Explanation:

It has been given that
`csc(\theta)=(1)/(sin\theta )` and we have to find the domain of the given expression.

As we know
`(1)/(sin\theta )` is defined only when
`sin\theta>0` or
`sin\theta<0` but not equal to zero.

Since sine of
`0, \pi, 2\pi......` are always zero. Therefore domain of
`(1)/(sin\theta )` is
`{\theta}<\pi or {\theta}>\pi`

That means Option D. All real numbers except multiples of pi is the answer.