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What is the domain of validity for csc theta =

/ 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.

Verified Answer :

RedRicin

For the three basic trig identities (sin, cos, tan) there are three more which act as their reciprocals (csc, sec, and cot respectively)

A reciprocal of x can be represented as 1/x.
Therefore, cscӨ can also be represented as the reciprocal of sinӨ...1/sinӨ.

In that case, our answer should always be true so long as we put in a real number for theta, because that's the domain of sinӨ, right? However, we also have to satisfy the domain of cscӨ, and the limitations become extremely obvious when you look at this reciprocal identity equation...sinӨ cannot be zero because it is impposible to divide by zero! Looking at the unit circle, any multiple of π will make sinӨ = 0, so there's your answer.
D. All real numbers except multiples of pi

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