Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Use the appropriate reciprocal identity to find each firnction value. Rarionalize denominators when applicable. $$\sec \theta, \text { given that } \cos \theta=\frac{5}{8}$$

Step-by-Step Solution

Answer

\(\sec \theta = \frac{8}{5}\).

Step 1: Understanding the Reciprocal Identity

The reciprocal identity states that the secant function, \(\sec \theta\), is the reciprocal of the cosine function, \(\cos \theta\). This means that \(\sec \theta = \frac{1}{\cos \theta}\).

Step 2: Substitute the Given Value

We are given \(\cos \theta = \frac{5}{8}\). Substitute this value into the reciprocal identity formula: \(\sec \theta = \frac{1}{\frac{5}{8}}\).

Step 3: Simplify the Expression

To simplify \(\frac{1}{\frac{5}{8}}\), use the property of division of fractions, which is to multiply by the reciprocal. Thus, \(\sec \theta = 1 \times \frac{8}{5} = \frac{8}{5}\).

Step 4: Rarionalize the Denominator if Necessary

Since \(\frac{8}{5}\) does not contain a radical in the denominator, no further rationalization is necessary. The expression is already in its simplest rational form.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is a trigonometric function. It is directly related to the cosine function. Specifically, it is the reciprocal of the cosine function. This relationship can be expressed through the formula:

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

Understanding this relationship is key to solving problems involving secant. When given \( \cos \theta \), simply take the reciprocal to find \( \sec \theta \). This means flipping the cosine fraction upside down. For example, if the cosine function value is \( \frac{5}{8} \), then the secant function value becomes \( \frac{8}{5} \). This reciprocal identity helps simplify computations and find the secant value efficiently.The secant function is particularly useful in various mathematical applications, including solving trigonometric equations and analyzing periodic functions. Remembering its connection to cosine will help streamline many mathematical processes.

Cosine Function

The cosine function, represented as \( \cos \theta \), is one of the primary trigonometric functions. It is defined in terms of a right-angled triangle as the ratio of the length of the adjacent side to the hypotenuse. In the unit circle, it is depicted as the horizontal coordinate of the angle \( \theta \).The cosine function is known for its periodic nature, repeating its values every \( 360^{\circ} \) or \( 2\pi \) radians. It takes values between -1 and 1. The reciprocal and periodic properties of cosine make it valuable when calculating transformations and analyzing waves.

• For example, if \( \cos \theta = \frac{5}{8} \), the angle \( \theta \) has its cosine value as the fraction \( \frac{5}{8} \).
• This indicates the proportionate relationship in a specific scenario or geometric configuration.

Cosine is foundational for solving many mathematical problems, especially those involving angles.

Division of Fractions

Division of fractions involves a unique process: multiplying by the reciprocal of the divisor. This is a straightforward rule, but it can sometimes cause confusion. To divide one fraction by another, you invert (flip) the second fraction and multiply.

• For example, dividing \( \frac{1}{\frac{5}{8}} \) requires you to multiply \( 1 \) by the reciprocal of \( \frac{5}{8} \), which is \( \frac{8}{5} \).
• The result is \( \frac{8}{5} \).

This technique simplifies the process and avoids direct division, which can be more complex.When dealing with trigonometric functions like secant, understanding this division process is crucial. It allows you to simplify the reciprocal identity efficiently. By following the steps of dividing and multiplying by reciprocals, you achieve the result faster and accurately. Proper application of division of fractions can greatly enhance your mathematical problem-solving abilities.