Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\operatorname{arcsec} \frac{2 \sqrt{3}}{3}$$

Step-by-Step Solution

Answer

The exact value of \(y\) is \(\frac{\pi}{6}\).

Step 1: Understand the Problem

We need to find the exact value of \(y\) such that \(y = \operatorname{arcsec} \frac{2\sqrt{3}}{3}\). This means we are looking for an angle \(y\) in radians whose secant value is \(\frac{2\sqrt{3}}{3}\).

Step 2: Recall the Definition of Arcsecant

Arcsecant or \(\operatorname{arcsec}(x)\) is the inverse function of secant. Thus, \(y = \operatorname{arcsec}\left(\frac{2\sqrt{3}}{3}\right)\) implies \(\sec(y) = \frac{2\sqrt{3}}{3}\). Since we're working within arcs,\(0 \leq y < \pi, y eq \frac{\pi}{2}\).

Step 3: Solve for Cosine

Remember that \(\sec(y) = \frac{1}{\cos(y)}\). Therefore, \(\cos(y) = \frac{3}{2\sqrt{3}}\). Simplify this by rationalizing the denominator: \(\cos(y) = \frac{3}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{2}\).

Step 4: Find y that satisfies \(\cos(y) = \frac{\sqrt{3}}{2}\)

The angle \(y\) where \(\cos(y) = \frac{\sqrt{3}}{2}\) is commonly known to be \(y = \frac{\pi}{6}\), since cosine is positive in the first quadrant and this is one of the special angles.

Step 5: Verify the Interval Condition

Check if \(y = \frac{\pi}{6}\) is within the valid range for arcsec, which is \([0,\pi)\) excluding \(\frac{\pi}{2}\). Since \(\frac{\pi}{6}\) is within this range, it satisfies all conditions.

Key Concepts

Arcsecant

The arcsecant, often written as \( \operatorname{arcsec}(x) \), is an inverse trigonometric function. It's the inverse of the secant function. To put it simply, this function finds an angle \( y \) whose secant is \( x \). Secant is the reciprocal of the cosine function, thus involving arcsecant is directly linked to understanding these reciprocal relations.

• The function arcsecant is only defined for values of \( x \) where \(|x| \ge 1\). This is because secant values arise from reciprocals of cosine values, and cosine ranges only between \(-1\) and \(1\).
• When working with arcsecant, it's important to consider certain intervals. Typically, the principal values of arcsecant are taken in the interval \( [0, \pi) \), except for \( \pi/2 \).

Understanding arcsecant involves a firm grasp of how the inverse trigonometric functions swap axes from a result back to an angle. Thus \( y = \operatorname{arcsec} \left( \frac{2\sqrt{3}}{3} \right) \) signifies finding an angle whose secant (cosine's reciprocal) equals \( \frac{2\sqrt{3}}{3} \).

Secant Function

The secant function, represented by \( \sec(\theta) \), is the reciprocal of the cosine function. Mathematically, this relationship is shown as \( \sec(\theta) = \frac{1}{\cos(\theta)} \). This function is particularly useful when solving problems involving ratios that are not easily expressed through cosine itself.

• Since \( \cos(\theta) \) can range from \(-1\) to \(1\), \( \sec(\theta) \) will be undefined for \( \theta = \pi/2 + n\pi \) for any integer \( n \), because cosine at these points equals zero, rendering \( \sec(\theta) = \frac{1}{0} \).
• This function becomes more relevant in trigonometry especially when dealing with challenging angles, making inverse trigonometry functional globally.

In the original exercise, the target was to find the angle \( y \) with \( \sec(y) = \frac{2\sqrt{3}}{3} \). This reflects how secant can be applied to calculate angles using known values and relations derivable from the cosine.

Unit Circle

The unit circle is a fundamental tool in trigonometry used to define the trigonometric functions for all real angles. It's a circle centered at the origin \((0,0)\) with a radius of one unit. Its usage simplifies the understanding of functions such as sine, cosine, and therefore secant, as well as their inverses.

• The unit circle allows visualization of angle measures and their corresponding coordinates on the circle as cosine and sine values.
• For example, when determining \( \cos(y) = \frac{\sqrt{3}}{2} \), this comes from identifying points on the unit circle that project to the x-axis at \( \frac{\sqrt{3}}{2} \), notably at \( \frac{\pi}{6} \) radians.

Incorporating the unit circle is critical, especially for trigonometry students, as it provides an intuitive grasp of periodicity and angles relating to the functions involved in trigonometric equations and identities. It lets you see why certain angles give predictable function values.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables within their domain. They are powerful tools in simplifying expressions and solving equations, crucial in handling problems involving inverse functions like arcsecant.

• An important identity is \( \sec(\theta) = \frac{1}{\cos(\theta)} \), used to derive the value of \( y \) like in our problem.
• Other identities, such as Pythagorean identities, sum and difference formulas, and double angle formulas, also play foundational roles in understanding both trigonometric and inverse trigonometric functions.

The problem at hand utilized the identity converting secant to cosine, allowing students to use familiar cosine values to find angles configured in special triangles or the unit circle. Such identities ease the shift between trigonometric measures and angles, enhancing problem-solving capabilities.