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

Verified step by step guidance

1. Understand that \( y = \cot^{-1}(-1) \) means we are looking for an angle \( y \) such that \( \cot(y) = -1 \).
2. Recall that the cotangent function, \( \cot(\theta) \), is the reciprocal of the tangent function, \( \tan(\theta) \). Therefore, \( \cot(\theta) = \frac{1}{\tan(\theta)} \).
3. Recognize that \( \cot(\theta) = -1 \) implies \( \tan(\theta) = -1 \).
4. Identify the angles where \( \tan(\theta) = -1 \). These angles are typically found in the second and fourth quadrants, specifically at \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{7\pi}{4} \) within the interval \( [0, 2\pi) \).
5. Since the range of \( \cot^{-1}(x) \) is \( (0, \pi) \), select the angle \( \theta = \frac{3\pi}{4} \) as the solution for \( y = \cot^{-1}(-1) \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cot⁻¹, are used to find angles when given a trigonometric ratio. For cotangent, the function cot⁻¹(x) returns the angle whose cotangent is x. Understanding the range and properties of these functions is essential for solving problems involving inverse trigonometric values.

Cotangent Function

The cotangent function is defined as the ratio of the adjacent side to the opposite side in a right triangle, or as the reciprocal of the tangent function. It is periodic with a period of π, meaning that cot(θ) = cot(θ + nπ) for any integer n. Knowing the values of cotangent at key angles helps in determining the corresponding angles for inverse cotangent.

Quadrants and Angle Values

In trigonometry, angles are often analyzed in terms of their position in the four quadrants of the Cartesian plane. The cotangent function is positive in the first and third quadrants and negative in the second and fourth. For the equation y = cot⁻¹(―1), recognizing that cotangent is negative helps identify the specific angles in the second and fourth quadrants where this condition holds.