Find the exact value of each real number y. Do not use a calculator.
y = sec⁻¹ (―2)

Verified step by step guidance

1. Understand that \( y = \sec^{-1}(-2) \) means we are looking for an angle \( y \) such that \( \sec(y) = -2 \).
2. Recall that the secant function is the reciprocal of the cosine function, so \( \sec(y) = \frac{1}{\cos(y)} \). Therefore, \( \cos(y) = -\frac{1}{2} \).
3. Identify the range of the inverse secant function, which is \([0, \pi] \) excluding \( \frac{\pi}{2} \).
4. Determine the angle \( y \) within this range where \( \cos(y) = -\frac{1}{2} \).
5. Recognize that \( \cos(y) = -\frac{1}{2} \) corresponds to the angle \( y = \frac{2\pi}{3} \) in the specified range.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sec⁻¹ (arcsec), are used to find the angle whose secant is a given value. For example, if y = sec⁻¹(x), then sec(y) = x. Understanding how to interpret these functions is crucial for solving problems involving angles and their corresponding trigonometric ratios.

Secant Function

The secant function, defined as sec(θ) = 1/cos(θ), is the reciprocal of the cosine function. It is important to know that the secant function is defined for all angles where the cosine is not zero. The range of the secant function is limited to values less than or equal to -1 and greater than or equal to 1, which is essential when determining the possible outputs for inverse secant.

Domain and Range of Inverse Functions

The domain and range of inverse functions are critical for understanding their behavior. For sec⁻¹(x), the domain is x ≤ -1 or x ≥ 1, while the range is restricted to angles in the intervals [0, π/2) and (π/2, π]. This means that when solving for y = sec⁻¹(−2), we must ensure that the output angle falls within the defined range, which helps in identifying the correct angle corresponding to the given secant value.