Multiple Choice
Evaluate the expression.
0. Functions
Inverse Trigonometric Functions
2:57 minutesProblem 78
Textbook Question
Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.
Verified step by step guidance1
Understand that \( \csc^{-1}(x) \) is the inverse cosecant function, which gives the angle \( \theta \) such that \( \csc(\theta) = x \).
Recall that \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Therefore, \( \csc^{-1}(-1) \) means we are looking for an angle \( \theta \) where \( \sin(\theta) = -1 \).
The sine function \( \sin(\theta) \) equals \(-1\) at specific angles. Consider the unit circle: \( \sin(\theta) = -1 \) at \( \theta = \frac{3\pi}{2} \) (or \( 270^\circ \)).
Verify that \( \theta = \frac{3\pi}{2} \) is within the range of the inverse cosecant function. The principal range for \( \csc^{-1}(x) \) is \([-\frac{\pi}{2}, \frac{\pi}{2}] \) excluding \( 0 \), but for negative values, we consider angles in the third and fourth quadrants.
Conclude that the angle \( \theta = \frac{3\pi}{2} \) satisfies the condition \( \csc(\theta) = -1 \), and thus \( \csc^{-1}(-1) = \frac{3\pi}{2} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, such as arcsin, arccos, and arccsc, are the functions that reverse the action of the corresponding trigonometric functions. For example, if y = sin(x), then x = arcsin(y). These functions are defined for specific ranges to ensure they are one-to-one, allowing for unique outputs for each input.
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Cosecant Function
The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). It is important to note that csc(x) is undefined where sin(x) = 0. The cosecant function is particularly relevant when evaluating expressions involving inverse cosecant, such as csc^{-1}(-1).
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Range of Inverse Cosecant
The range of the inverse cosecant function, csc^{-1}(x), is limited to the intervals (-∞, -1] and [1, ∞). This means that csc^{-1}(x) can only yield values outside the interval (-1, 1), which is crucial when evaluating expressions like csc^{-1}(-1), as it indicates the specific angle whose cosecant is -1.
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