Final answer:
The output intervals for the inverse trigonometric functions are: for y=sin−1 x it is [-π/2, π/2], for y=cos−1 x it is [0, π], and for y=tan−1 x it is (-π/2, π/2) (not including π/2). These intervals ensure that the functions have a single output for any given input.
Step-by-step explanation:
For each inverse trigonometric function, there is an interval in which the output must fall to be considered valid. This restriction ensures that each function has only one output for a given input, which is necessary since functions in mathematics are defined to have exactly one output for each input.
- Sine inverse (y=sin−1 x): The output y must be in the interval [-π/2, π/2], which corresponds to angles that lie in the first and fourth quadrants where the sine function is increasing.
- Cosine inverse (y=cos−1 x): The output y must be in the interval [0, π], covering angles from the first and second quadrants where the cosine function is decreasing.
- Tangent inverse (y=tan−1 x): The output y must be in the interval [ -π/2, π/2 ] but not including π/2 itself, capturing the angles where the tangent function is increasing from negative infinity to positive infinity.
These intervals are chosen based on the periodic nature of trigonometric functions and their graphs, which oscillate and repeat over intervals. For sine and cosine functions, the principal range is chosen so that every possible output value corresponds to exactly one angle in the interval, preserving the function property. The same logic applies to the tangent function, even though its graph has vertical asymptotes and repeats every π radians.