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

1. Understand that \( y = \text{arcsec}(x) \) means \( \sec(y) = x \).
2. Set up the equation \( \sec(y) = \frac{2\sqrt{3}}{3} \).
3. Recall that \( \sec(y) = \frac{1}{\cos(y)} \), so \( \cos(y) = \frac{3}{2\sqrt{3}} \).
4. Simplify \( \cos(y) = \frac{3}{2\sqrt{3}} \) to \( \cos(y) = \frac{\sqrt{3}}{2} \) by rationalizing the denominator.
5. Determine the angle \( y \) where \( \cos(y) = \frac{\sqrt{3}}{2} \) within the range \( [0, \pi] \) for arcsec.

Key Concepts

Arcsecant Function

The arcsecant function, denoted as arcsec(x), is the inverse of the secant function. It is defined for values of x where |x| ≥ 1, and it returns an angle whose secant is x. Understanding this function is crucial for solving problems involving angles and their corresponding secant values.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. In the context of the secant function, sec(θ) is defined as the ratio of the hypotenuse to the adjacent side in a right triangle. Knowing how to manipulate these ratios is essential for finding exact values of angles.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to specific angles where the sine, cosine, and tangent values can be expressed as simple fractions or radicals. For example, angles like 30°, 45°, and 60° have well-known exact values. Recognizing these angles helps in determining the exact value of y in the given equation without the use of a calculator.