In Exercises 1–26, find the exact value of each expression._csc⁻¹ (− 2√3/3)

Verified step by step guidance

1. Understand that \( \csc^{-1}(x) \) is the inverse cosecant function, which means we are looking for an angle \( \theta \) such that \( \csc(\theta) = x \).
2. Recall that \( \csc(\theta) = \frac{1}{\sin(\theta)} \), so we need to find \( \theta \) such that \( \sin(\theta) = -\frac{3}{2\sqrt{3}} \).
3. Simplify \( \sin(\theta) = -\frac{3}{2\sqrt{3}} \) to \( \sin(\theta) = -\frac{\sqrt{3}}{2} \) by rationalizing the denominator.
4. Identify the angle \( \theta \) in the unit circle where \( \sin(\theta) = -\frac{\sqrt{3}}{2} \). This occurs at specific angles in the third and fourth quadrants.
5. Determine the exact angle(s) \( \theta \) that satisfy the condition, keeping in mind the range of \( \csc^{-1}(x) \), which is \([-\frac{\pi}{2}, \frac{\pi}{2}] \) excluding zero.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as csc⁻¹ (cosecant inverse), are used to find angles when given a trigonometric ratio. For example, csc⁻¹(x) gives the angle whose cosecant is x. Understanding how to interpret these functions is crucial for solving problems involving angles and their corresponding ratios.

Cosecant Function

The cosecant function is the reciprocal of the sine function, defined as csc(θ) = 1/sin(θ). This means that if you know the value of csc(θ), you can find sin(θ) by taking the reciprocal. Recognizing the relationship between sine and cosecant is essential for evaluating expressions involving these functions.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions are specific values that correspond to well-known angles, typically found on the unit circle. For instance, angles like 30°, 45°, and 60° have exact sine, cosine, and tangent values. Knowing these exact values helps in determining the angles associated with trigonometric ratios, especially when dealing with inverse functions.