Final answer:
The sine and cosine functions have restricted domains to ensure they have invertible functions, limited to ranges where they are strictly increasing or decreasing. Their reciprocal functions, cosecant and secant, have slightly different domains because they are undefined at points where sine and cosine are zero.
Step-by-step explanation:
The domains for the sine and cosine functions are restricted to ensure that they have inverses which are also functions. The sine function oscillates between +1 and -1 and is periodic with a period of 2π radians. To restrict the domain for the function θ = sin-1(x) to ensure it is a function, we limit θ to values between -π/2 and π/2, where it is strictly increasing and passes the horizontal line test.
For the cosine function, the domain is similarly restricted. We limit the domain for θ = cos-1(x) to values between 0 and π because this section of the cosine curve represents a unique value of θ for each value of x, making the function invertible.
The secant and cosecant functions are reciprocals of cosine and sine respectively; however, their domains are not the same because the secant and cosecant functions are undefined where their corresponding functions are zero. The domain for cosecant, the inverse of sine, excludes values where sine is zero, such as multiples of π. Similarly, the domain for secant, the inverse of cosine, excludes values where cosine is zero, such as π/2 and 3π/2.