91Ó°ÊÓ

The secant inverse, denoted as \( \sec^{-1}(x) \), is a function that finds an angle whose secant value is \( x \). Understanding this requires knowing that secant is the reciprocal of cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \). The range for \( \sec^{-1}(x) \) is \([0, \pi] \), excluding \( \frac{\pi}{2} \), because \( \sec(\theta) \) can never be zero or undefined in this range. Let's evaluate \( \sec^{-1}(-3) \):- We look for an angle \( \theta \) such that \( \sec(\theta) = -3 \).- This implies \( \cos(\theta) = -\frac{1}{3} \), and \( \theta \) should be in the second quadrant where cosine values are negative.- Therefore, \( \sec^{-1}(-3) \) can be given as \( \cos^{-1}(-\frac{1}{3}) \), which yields the angle whose cosine is \(-\frac{1}{3}\) in the valid range.

The cosecant inverse function, \( \csc^{-1}(x) \), finds an angle for which the cosecant is \( x \). Cosecant is the reciprocal of sine, meaning \( \csc(\theta) = \frac{1}{\sin(\theta)} \). The range for \( \csc^{-1}(x) \) is \([-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}] \), which helps to determine if the angle lies in the first or the fourth quadrants. To find \( \csc^{-1}(1.7) \):- Since \( 1.7 > 1 \), \( \theta \) should be in the first quadrant, where both sine and cosecant are positive.- We calculate \( \sin(\theta) = \frac{1}{1.7} \), which means \( \theta = \sin^{-1}(\frac{1}{1.7}) \).- Thus, \( \csc^{-1}(1.7) \) results in the angle whose sine is \( \frac{1}{1.7} \), fitting in the specified range.

The cotangent inverse, expressed as \( \cot^{-1}(x) \), provides an angle whose cotangent value is \( x \). Cotangent relates to tangent as \( \cot(\theta) = \frac{1}{\tan(\theta)} \). The function's range is \((0, \pi) \), helping identify the correct quadrant for the angle's position.For \( \cot^{-1}(-2) \):- We need an angle \( \theta \) where \( \cot(\theta) = -2 \), meaning \( \tan(\theta) = -\frac{1}{2} \).- This places \( \theta \) in the second quadrant, where tangent is negative and cotangent follows suit.- Computation gives \( \theta = \pi - \tan^{-1}(\frac{1}{2}) \), representing \( \cot^{-1}(-2) \), where the angle resolves to balancing \( -\frac{1}{2} \) in the allowed range.