Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, such as tan⁻¹ (arctan), are used to find angles when given a ratio of sides in a right triangle. For example, tan⁻¹(x) gives the angle whose tangent is x. Understanding these functions is crucial for solving problems that require determining angles from known ratios.
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Tangent Function
The tangent function, defined as the ratio of the opposite side to the adjacent side in a right triangle, is periodic and has a range of all real numbers. The value of tan(θ) can be found for various angles, and knowing that tan(45°) = 1 helps in determining the angle when tan⁻¹(1) is evaluated.
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Principal Value of Inverse Functions
The principal value of an inverse function refers to the specific output range that the function is defined to return. For tan⁻¹(x), the principal value is restricted to the interval (-π/2, π/2). This means that when finding y = tan⁻¹(1), the solution will be the angle within this range where the tangent equals 1.
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