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

Verified step by step guidance

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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Key Concepts

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

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.

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