Final answer:
The domains for sine and cosine are all real numbers due to their relation to circular motion, and they oscillate between -1 and +1. Cosecant and secant have restricted domains, excluding angles where their reciprocal functions sine and cosine are zero.
Step-by-step explanation:
The domains for sine and cosine are all real numbers because these functions relate to circular motion, where the angle of rotation can be any real number. The sine and cosine values oscillate between -1 and +1. For any angle θ, the sine function can be given by sin θ = opposite/hypotenuse, and the cosine function by cos θ = adjacent/hypotenuse, as for a right triangle.
The cosecant and secant are the reciprocals of the sine and cosine functions, respectively. However, the domains for cosecant and secant do not include the angles where sine and cosine are zero – specifically, multiples of π for sine and multiples of π/2 for cosine – because the values of cosecant and secant approach infinity as sine and cosine approach zero. This difference in domains is due to the fact that a function and its reciprocal undefined when the original function is zero.