The point (π/4, 1) lies on the graph of y = tan x. Therefore, the point _______ lies on the graph of y = tan⁻¹ x.

Verified step by step guidance

1. Identify the given point (\(\frac{\pi}{4}, 1\)) on the graph of \(y = \tan x\).
2. Understand that the point (\(\frac{\pi}{4}, 1\)) means that when \(x = \frac{\pi}{4}\), \(y = \tan(\frac{\pi}{4}) = 1\).
3. Recognize that the inverse function \(y = \tan^{-1} x\) swaps the roles of \(x\) and \(y\).
4. Therefore, the point on the graph of \(y = \tan^{-1} x\) will be (1, \(\frac{\pi}{4}\)).
5. Conclude that the point (1, \(\frac{\pi}{4}\)) lies on the graph of \(y = \tan^{-1} x\).

Key Concepts

Tangent Function

The tangent function, denoted as y = tan x, is a fundamental trigonometric function that relates the angle x in a right triangle to the ratio of the opposite side to the adjacent side. It is periodic with a period of π, meaning it repeats its values every π radians. The function is undefined at odd multiples of π/2, where the cosine value is zero.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as y = tan⁻¹ x, are used to find the angle whose tangent is a given number. The range of the inverse tangent function is limited to (-π/2, π/2), which means it only returns angles in the first and fourth quadrants. This is crucial for determining the corresponding angle for a given tangent value.

Coordinate Transformation

In the context of trigonometric functions, coordinate transformation involves switching between the input-output pairs of a function and its inverse. For the point (π/4, 1) on the graph of y = tan x, the corresponding point on the graph of y = tan⁻¹ x can be found by swapping the x and y coordinates, resulting in the point (1, π/4). This reflects the relationship between a function and its inverse.