which trigonometric function requires a domain restriction of [-pi/2 pi/2] to make it invertible
The trigonometric function that requires a domain restriction of [-π/2, π/2] to make it invertible is the tangent function, commonly denoted as tan(x).
The trigonometric function that requires a domain restriction of [-π/2, π/2] to make it invertible is the tangent function (tan(x)).
To understand why the domain restriction is necessary, let's consider the graph of the tangent function. The tangent function is defined as the ratio of the sine function to the cosine function: tan(x) = sin(x) / cos(x).
If we don't apply any domain restriction, the tangent function repeats its values every π radians (180 degrees) as the cosine function crosses zero. This repetition makes it impossible for the function to have a unique inverse.
However, by restricting the domain to the interval [-π/2, π/2], we eliminate the repeated values and ensure that the tangent function has a one-to-one mapping between the given interval and its range.
In other words, the domain restriction limits the tangent function to only a single period, resulting in an invertible function within the given interval.