Find the exact value of each real number y if it exists. Do not use a calculator.
y = arcsec (2√3)/3

검증된 단계별 안내

Recall the definition of the arcsecant function: \(y = \arcsec(x)\) means \(\sec(y) = x\) and \(y\) lies in the domain of the arcsec function, typically \([0, \pi]\) excluding \(\frac{\pi}{2}\).

Set up the equation from the problem: \(\sec(y) = \frac{2\sqrt{3}}{3}\).

Use the identity \(\sec(y) = \frac{1}{\cos(y)}\) to rewrite the equation as \(\frac{1}{\cos(y)} = \frac{2\sqrt{3}}{3}\).

Solve for \(\cos(y)\) by taking the reciprocal: \(\cos(y) = \frac{3}{2\sqrt{3}}\).

Simplify \(\cos(y)\) and then determine the angle \(y\) in the interval \([0, \pi]\) (excluding \(\frac{\pi}{2}\)) whose cosine matches this value, using known special angles and exact values.

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주요 개념

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Definition of the Arcsecant Function

The arcsecant function, denoted as arcsec(x), is the inverse of the secant function. It returns the angle whose secant is x. Since sec(θ) = 1/cos(θ), arcsec(x) finds θ such that sec(θ) = x, with θ typically in the range [0, π] excluding π/2.

Relationship Between Secant and Cosine

Secant is the reciprocal of cosine, so sec(θ) = 1/cos(θ). To find an angle from a secant value, convert it to cosine by cos(θ) = 1/sec(θ). This relationship helps in identifying the angle by comparing cosine values to known special angles.

Exact Values of Trigonometric Functions for Special Angles

Certain angles like π/6, π/4, and π/3 have well-known exact trigonometric values involving √2 and √3. Recognizing these values allows one to find exact angles without a calculator by matching the given secant value to the reciprocal of a known cosine value.

Find the exact value of each real number y if it exists. Do not use a calculator.y = arcsec (2√3)/3