Problem 21 - A Graphical Approach to Precalculus with Limits

Question

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$sec $30^{\circ}$$

Step-by-Step Solution

Answer

The exact value is \( \frac{2\sqrt{3}}{3} \); the decimal approximation is 1.155.

Step 1: Recall the Secant Function Definition

The secant function, \( \sec \theta \), is defined as the reciprocal of the cosine function. Therefore, \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Determine Cosine of the Angle

For \( 30^{\circ} \), we know from trigonometric values of special angles that \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \).

Step 3: Calculate the Exact Value of Secant

Using the definition of secant, \( \sec 30^{\circ} = \frac{1}{\cos 30^{\circ}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \). To rationalize the denominator, multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \) to get \( \sec 30^{\circ} = \frac{2\sqrt{3}}{3} \).

Step 4: Check for Rational or Irrational Result

The exact value \( \frac{2\sqrt{3}}{3} \) contains \( \sqrt{3} \), which is irrational. Therefore, the exact value itself is irrational.

Step 5: Find the Decimal Approximation

Use a calculator to approximate \( \sqrt{3} \approx 1.732 \). Then \( \frac{2\sqrt{3}}{3} \approx \frac{2 \times 1.732}{3} \approx 1.155 \).

Key Concepts

Secant Function

The secant function is a fascinating and fundamental part of trigonometry. It is closely related to the cosine function and provides valuable insights into the nature of angles. The secant function, denoted as \( \sec \theta \), is defined as the reciprocal of the cosine function. This means that \( \sec \theta = \frac{1}{\cos \theta} \). In simpler terms, if you know the value of cosine at a particular angle, you can easily find the secant by taking the reciprocal of that value.

Because the secant is the reciprocal of cosine, it behaves similarly in terms of its range and domain. Whenever the cosine value is zero, the secant function will be undefined, as dividing by zero is not possible. This typically occurs at angles such as \( 90^{\circ} \) and \( 270^{\circ} \), where the cosine of these angles is zero. This characteristic makes the secant function an essential tool for understanding and calculating values for a variety of trigonometric scenarios.

Understanding the secant function can also help in grasping why certain angles behave uniquely in trigonometric calculations and why secant values can sometimes become very large or undefined. It's always important to remember that the secant function is specifically dependent on the cosine function for its calculations.

Special Angles

Special angles in trigonometry are particularly important because their trigonometric values are well-known and widely used in both theoretical and practical applications. These angles include \( 0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \) and \( 90^{\circ} \), among others. These specific angles are known as 'special' because their sine, cosine, and tangent values are either integers or simple fractions involving square roots.

For example, at \( 30^{\circ} \), the cosine value is \( \frac{\sqrt{3}}{2} \), making it particularly easy to work with in problems involving ratios of right triangles in geometry. What's fascinating is these values remain constant and are widely used across various fields such as physics, engineering, and even computer graphics.

By using these pre-determined values for the special angles, we can avoid lengthy calculations and perform complex computations swiftly. The wholesale memorization of these values is often recommended because they reduce potential calculation errors and significantly streamline the problem-solving process. This knowledge helps in comprehending the deeper relationships between different trigonometric functions.

Decimal Approximation

Decimal approximation is a valuable tool in mathematics, especially when dealing with irrational numbers. It's a process used to approximate irrational numbers, which cannot be exactly expressed as simple fractions or decimals. Irrational numbers have non-repeating and non-terminating decimal expansions.

For instance, \( \sqrt{3} \) is an irrational number with a non-repeating decimal sequence. In many practical situations, the exact irrational value is not necessary. Instead, a decimal approximation is sufficient for calculations. Using a calculator, \( \sqrt{3} \) is typically approximated as \( 1.732 \).

This approximation becomes crucial when we compute trigonometric function values like \( \sec 30^{\circ} \). After determining the exact value of \( \frac{2\sqrt{3}}{3} \), approximating this gives \( 1.155 \).

Decimal approximations provide a convenient and accessible way to express and work with values that would otherwise be cumbersome in their exact irrational form. They simplify mathematical tasks, making it possible to perform meaningful real-world calculations without the need for high precision.