Find the exact value of each real number y if it exists. Do not use a calculator.
y = csc⁻¹ √2/2

Verified step by step guidance

Recall that the function \( y = \csc^{-1}(x) \) means \( \csc(y) = x \), where \( y \) is the angle whose cosecant is \( x \).

Given \( y = \csc^{-1}\left( \frac{\sqrt{2}}{2} \right) \), rewrite this as \( \csc(y) = \frac{\sqrt{2}}{2} \).

Since \( \csc(y) = \frac{1}{\sin(y)} \), set up the equation \( \frac{1}{\sin(y)} = \frac{\sqrt{2}}{2} \).

Solve for \( \sin(y) \) by taking the reciprocal of both sides: \( \sin(y) = \frac{2}{\sqrt{2}} \).

Simplify \( \sin(y) \) and then determine all possible angles \( y \) within the domain of \( \csc^{-1} \) that satisfy this sine value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Cosecant Function (csc⁻¹)

The inverse cosecant function, csc⁻¹(x), returns the angle whose cosecant is x. Since cosecant is the reciprocal of sine, csc⁻¹(x) = θ means sin(θ) = 1/x. Understanding this relationship helps convert the problem into finding an angle with a known sine value.

Reciprocal Relationship Between Sine and Cosecant

Cosecant is defined as csc(θ) = 1/sin(θ). This reciprocal relationship allows us to rewrite expressions involving csc⁻¹ in terms of sine, simplifying the problem to finding an angle with a specific sine value, which is often easier to evaluate using known unit circle values.

Unit Circle Values and Exact Trigonometric Ratios

Familiarity with exact sine values of common angles (like 30°, 45°, 60° or π/6, π/4, π/3) is essential. Since the problem requires no calculator, recognizing that sin(π/4) = √2/2 helps identify the angle corresponding to the given value, enabling the exact evaluation of the inverse cosecant.

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