Textbook Question
Find the exact value of each real number y if it exists. Do not use a calculator.
y = arccos (―√3/2)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 31
Textbook Question
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 guidance1
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.
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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.
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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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