Final answer:
Restricted domains for one-to-one trigonometric functions are as follows: sine and tangent have a restricted domain of (-π/2, π/2), cosine and secant have [0, π], cotangent has (0, π). These intervals prevent the functions from repeating values, thus maintaining their one-to-one property.
Step-by-step explanation:
To give restricted domains for trigonometric functions that ensure they are one-to-one functions, we must consider their periodic nature and the fact that they repeat their values over specific intervals. For one-to-one functions, we need each output value to correspond to exactly one input value, which means we must restrict the domain to an interval where the function's graph doesn't double back on itself.
For sin and cos, the commonly accepted restricted domains are (-π/2, π/2) and [0, π], respectively. This is because sine is one-to-one on the interval (-π/2, π/2), where it increases from -1 to 1, and cosine is one-to-one on the interval [0, π], where it decreases from 1 to -1.
For the tan function, the restricted domain is also (-π/2, π/2), excluding the endpoints where tan is undefined due to division by zero in the sine over cosine definition. For cot, the restricted domain is (0, π), for similar reasons to tan.
When it comes to sec and csc, their restricted domains relate to those of cos and sin, respectively. Therefore, the restricted domain for sec is [0, π] excluding the point where cos is zero, and the restricted domain for csc is (-π/2, π/2) also excluding the points where sin is zero.