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

Verified step by step guidance

1. Recall that the function \( \csc^{-1}(x) \) is the inverse cosecant function, which gives an angle \( \theta \) such that \( \csc(\theta) = x \). Our goal is to find \( \theta \) where \( \csc(\theta) = -\frac{2\sqrt{3}}{3} \).
2. Use the identity relating cosecant and sine: \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Therefore, \( \sin(\theta) = \frac{1}{\csc(\theta)} = \frac{1}{-\frac{2\sqrt{3}}{3}} = -\frac{3}{2\sqrt{3}} \).
3. Simplify the expression for \( \sin(\theta) \) by rationalizing the denominator: multiply numerator and denominator by \( \sqrt{3} \) to get \( \sin(\theta) = -\frac{3\sqrt{3}}{2 \times 3} = -\frac{\sqrt{3}}{2} \).
4. Determine the angle \( \theta \) whose sine is \( -\frac{\sqrt{3}}{2} \). Recall that \( \sin(\theta) = \pm \frac{\sqrt{3}}{2} \) corresponds to reference angles of \( \frac{\pi}{3} \) (or 60 degrees). Since the sine is negative, \( \theta \) must be in either the third or fourth quadrant.
5. Identify the principal value range for \( \csc^{-1}(x) \), which is usually \( [-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}] \) excluding zero. Find the angle \( \theta \) in this range with \( \sin(\theta) = -\frac{\sqrt{3}}{2} \).

Key Concepts

Inverse Cosecant Function (csc⁻¹)

The inverse cosecant function, csc⁻¹(x), returns the angle whose cosecant is x. It is the inverse of the cosecant function, which is defined as csc(θ) = 1/sin(θ). Understanding its domain and range is essential to find the correct angle corresponding to a given value.

Relationship Between Cosecant and Sine

Cosecant is the reciprocal of sine, so csc(θ) = 1/sin(θ). To find an angle from a cosecant value, first find the sine value by taking the reciprocal. This relationship helps convert the problem into finding an angle from a sine value, which is more straightforward.

Exact Values of Special Angles

Certain angles have well-known exact sine and cosecant values, often involving √2, √3, and rational numbers. Recognizing these special angles (like 30°, 45°, 60° or π/6, π/4, π/3) allows you to identify the angle corresponding to the given cosecant value without a calculator.