91Ó°ÊÓ

In trigonometry, the inverse secant function, denoted as \(\sec^{-1}(x)\), is quite unique. This function helps us find the angle \(\theta\) whose secant is \(x\). Now, secant is the reciprocal of the cosine function. This means if \(\sec \theta = x\), then \(\cos \theta = \frac{1}{x}\).

Let's talk about the domain and range of \(\sec^{-1}(x)\). The domain is where the secant function takes on values, specifically \(|x| \geq 1\). The range is typically \([0, \pi]\) excluding \(\frac{\pi}{2}\) because at that angle, secant is undefined. Why? Because cosine equals zero at \(\frac{\pi}{2}\), and division by zero is not allowed.

The exercise asks for \(\sec^{-1}(-3)\). Since \(-3\) is in the domain (\(-3 < -1\)), it's within the working range. We need to find an angle \(\theta\) in the second quadrant, given secant is negative there. This ensures our cosine, \(-\frac{1}{3}\), is correctly from the second quadrant. Using trigonometric tables or a unit circle, we identify \(\theta\) to be approximately \(1.905\) radians.

The inverse cosecant function, \(\csc^{-1}(x)\), aids us in finding an angle \(\theta\) such that the cosecant of that angle is \(x\). Cosecant is the reciprocal of the sine function, meaning if \(\csc \theta = x\), then \(\sin \theta = \frac{1}{x}\).

For the domain of \(\csc^{-1}(x)\), values must meet \(|x| \geq 1\). Furthermore, its range spreads between \([\frac{-\pi}{2}, \frac{\pi}{2}]\) but excludes \(0\) as sine cannot be zero here (cosecant becomes undefined because of division by zero).

To find \(\csc^{-1}(1.7)\), first assure yourself it's in the domain, which it is because \(1.7 > 1\). You then compute \(\sin \theta = \frac{1}{1.7}\). Using these values in a calculator gives us \(\theta \approx 0.619\) radians.

The inverse cotangent function, or \(\cot^{-1}(x)\), pinpoints the angle \(\theta\) for which the cotangent equals \(x\). Since cotangent is the reciprocal of the tangent function, if \(\cot \theta = x\), then \(\tan \theta = \frac{1}{x}\). The range of this inverse function is usually set as \((0, \pi)\), which are angles where cotangent can maintain all real values.

The scenario with \(\cot^{-1}(-2)\) involves finding an angle \(\theta\) where the cotangent is \(-2\), or equivalently, the tangent is \(-\frac{1}{2}\). Tangent becomes negative in the second quadrant, ensuring that \(\theta\) is in the correct range. By using a calculator or reference tables, \(\theta\) is approximately \(2.677\) radians.