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

Guida verificata passo dopo passo

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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Concetti chiave

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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.

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6:22

Graphs of Secant and Cosecant Functions

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.

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Percorso guidato

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Graphs of Secant and Cosecant Functions

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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