Find the exact value of each real number y if it exists. Do not use a calculator.
y = tan⁻¹ 1

Verified step by step guidance

1. Recognize that \( y = \tan^{-1}(1) \) means we are looking for an angle \( y \) such that \( \tan(y) = 1 \).
2. Recall that the tangent function \( \tan(\theta) \) is equal to 1 at specific angles. In the unit circle, this occurs where the sine and cosine values are equal.
3. Identify the angles in the unit circle where \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = 1 \). This happens when \( \sin(\theta) = \cos(\theta) \).
4. The most common angle where \( \sin(\theta) = \cos(\theta) \) is \( \frac{\pi}{4} \) radians or 45 degrees, as both sine and cosine are \( \frac{\sqrt{2}}{2} \) at this angle.
5. Conclude that \( y = \frac{\pi}{4} \) is the principal value for \( \tan^{-1}(1) \), as the inverse tangent function typically returns values in the range \(-\frac{\pi}{2}, \frac{\pi}{2}\).

Key Concepts

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.

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.

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.