Problem 185 - Precalculus

Question

For the following exercises, find the exact value of each expression. $$ \sec \frac{\pi}{6} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Definition of Secant

The secant function is the reciprocal of the cosine function. Therefore, \( \sec \theta = \frac{1}{\cos \theta} \). To find \( \sec \frac{\pi}{6} \), we first need to find \( \cos \frac{\pi}{6} \).

Step 2: Recall the Cosine Value

The cosine of \( \frac{\pi}{6} \) is a standard trigonometric value. \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \).

Step 3: Find the Secant Value

Using the reciprocal identity, \( \sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}} \). Substitute the cosine value from step 2: \( \sec \frac{\pi}{6} = \frac{1}{\frac{\sqrt{3}}{2}} \).

Step 4: Simplify the Expression

To simplify \( \frac{1}{\frac{\sqrt{3}}{2}} \), multiply the numerator by the reciprocal of the denominator: \( \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \).

Step 5: Rationalize the Denominator

Multiply the numerator and the denominator by \( \sqrt{3} \) to eliminate the square root from the denominator: \( \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \).

Step 6: Verify the Result

We've verified that \( \sec \frac{\pi}{6} = \frac{2\sqrt{3}}{3} \). All simplifications have been confirmed correct.

Key Concepts

Secant Function

The secant function is one of the six main trigonometric functions. It is denoted as \( \sec \theta \) and has a direct relationship with the cosine function. Specifically, the secant function is defined as the reciprocal of the cosine function.

Here's a closer look at how it's defined:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This means that to find the secant of an angle, you first need to determine its cosine value and then take the reciprocal. This relationship is foundational and often simplifies the process of working with trigonometric expressions. The secant function, like its counterparts, is periodical and has specific values for standard angles like \( \frac{\pi}{6} \), which can be found using its reciprocal identity.

Reciprocal Identity

The reciprocal identity is a crucial concept in trigonometry. It refers to how certain trigonometric functions can be expressed as the inverse, or reciprocal, of one another. In the case of our exercise, we are exploring the secant function, which is the reciprocal of the cosine function.

Here is how the reciprocal identities work for secant:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• This is applicable for all angles except for those where cosine results in zero, as this would make secant undefined.

These identities are immensely helpful when solving trigonometric equations and enable us to convert between different functions for simplification and calculation.

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate radicals, such as square roots, from the denominator of a fraction. It is a preferred mathematical practice since it often simplifies further calculations.

To rationalize the denominator, do the following steps:

• Multiply the numerator and the denominator by the same radical present in the denominator.
• This process will remove the radical from the denominator by utilizing the property \( \sqrt{a} \times \sqrt{a} = a \).

In our example, \( \frac{2}{\sqrt{3}} \) was rationalized by multiplying both the numerator and the denominator by \( \sqrt{3} \), resulting in \( \frac{2\sqrt{3}}{3} \). This method is not only systematic but essential for presenting final answers in the most accepted format.