Problem 126 - Calculus Volume 1

Question

Evaluate the following functional values. $$\sec \left(\frac{\pi}{6}\right)$$

Step-by-Step Solution

Answer

\(\sec\left(\frac{\pi}{6}\right) = \frac{2\sqrt{3}}{3}\).

Step 1: Identify the Related Trigonometric Function

The function \(\sec(x)\) is the reciprocal of the cosine function, meaning \(\sec(x) = \frac{1}{\cos(x)}\). Therefore, to evaluate \(\sec\left(\frac{\pi}{6}\right)\), we first need to find \(\cos\left(\frac{\pi}{6}\right)\).

Step 2: Determine the Cosine Value for the Angle

The angle \(\frac{\pi}{6}\) is a commonly known angle in trigonometry. The cosine of \(\frac{\pi}{6}\) is \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\).

Step 3: Calculate the Secant Value

Now, use the cosine value to find the secant: \(\sec\left(\frac{\pi}{6}\right) = \frac{1}{\cos\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{\sqrt{3}}{2}}\).

Step 4: Simplify the Expression

To simplify \(\frac{1}{\frac{\sqrt{3}}{2}}\), multiply the numerator and the denominator by 2. This yields \(\sec\left(\frac{\pi}{6}\right) = \frac{2}{\sqrt{3}}\).

Step 5: Rationalize the Denominator

Multiply the numerator and the denominator of \(\frac{2}{\sqrt{3}}\) by \(\sqrt{3}\) to rationalize the denominator. This gives \(\frac{2\sqrt{3}}{3}\).

Key Concepts

Secant Function

The secant function, often written as \( \sec(x) \), is one of the six fundamental trigonometric functions. Secant is defined as the reciprocal of the cosine function. So, \( \sec(x) = \frac{1}{\cos(x)} \). This means that when you have a secant value, you're essentially flipping the cosine value for that angle. This makes secant particularly useful when working with angles or phases in various applications such as physics and engineering.

Understanding this reciprocal relationship helps tremendously when solving trigonometric equations because it connects directly back to the more familiar cosine function. If you ever need to convert between secant and other trigonometric identities, remembering this reciprocal connection can simplify the process significantly.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions extend beyond just the secant. They include cosecant and cotangent as well. These functions are as follows:

• Secant \((\sec(x)) = \frac{1}{\cos(x)}\)
• Cosecant \((\csc(x)) = \frac{1}{\sin(x)}\)
• Cotangent \((\cot(x)) = \frac{1}{\tan(x)}\)

These reciprocal functions are crucial in cases where division by zero might occur if directly using the sine, cosine, or tangent functions. They help transform complex trigonometric problems into more manageable forms.

Moreover, reciprocal trigonometric functions can simplify many integrals and derivatives in calculus. Thus, understanding how to manipulate these functions through their reciprocal relationships can help solve broader trigonometric identities and equations more fluidly.

Rationalizing the Denominator

Rationalizing the denominator is a mathematical process used to eliminate square roots or irrational numbers from the bottom of a fraction. In our example, after calculating \( \sec\left(\frac{\pi}{6}\right) \), we arrived at \( \frac{2}{\sqrt{3}} \).To rationalize this, multiply both numerator and denominator by \( \sqrt{3} \) to get \( \frac{2\sqrt{3}}{3} \).

This process makes expressions easier to understand and work with, especially when dealing with algebraic operations like addition and subtraction. Rationalizing the denominator is important because most mathematical conventions prefer fractions do not have radicals in the denominator.
By understanding and applying rationalization, one ensures the expression is simplified in its most conventional form, facilitating clearer interpretation and further manipulation of the expression.