Range of Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle measures when the ratios of sides of a right triangle are given. The range of inverse trigonometric functions depends on the range of the corresponding trigonometric functions. The range of the inverse trigonometric functions is the set of all possible values that their corresponding trigonometric functions can take.

Sin-1x

The inverse sine function, sin-1x, gives the angle whose sine is x. The range of the inverse sine function is [-π/2, π/2]. This is because the sine function is a periodic function with a period of 2π. Therefore, the sine function repeats itself every 2π. The range of the sine function is [-1, 1]. Therefore, the inverse sine function can take any value between -π/2 and π/2, inclusive, since the sine function is also continuous and one-to-one in this interval.

Sec-1x

The inverse secant function, sec-1x, gives the angle whose secant is x. The range of the inverse secant function is [0, π/2] ∪ [π, 3π/2]. This is because the secant function is a periodic function with a period of π. Therefore, the secant function repeats itself every π. The range of the secant function is (-∞, -1] ∪ [1, ∞). Therefore, the inverse secant function can take any value between 0 and π/2, inclusive, or any value between π and 3π/2, inclusive, since the secant function is also continuous and one-to-one in these intervals.

Tan-1x

The inverse tangent function, tan-1x, gives the angle whose tangent is x. The range of the inverse tangent function is (-π/2, π/2). This is because the tangent function is a periodic function with a period of π. Therefore, the tangent function repeats itself every π. The range of the tangent function is (-∞, ∞). Therefore, the inverse tangent function can take any value between -π/2 and π/2, exclusive, since the tangent function is also continuous and one-to-one in this interval.

Conclusion

In summary, the range of sin-1x is [-π/2, π/2], the range of sec-1x is [0, π/2] ∪ [π, 3π/2], and the range of tan-1x is (-π/2, π/2). It is important to note that the range of inverse trigonometric functions can vary depending on the domain of the function. Therefore, it is important to consider the domain of the function when determining its range.