Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Use the appropriate reciprocal identity to find each firnction value. Rarionalize denominators when applicable. $$\sec \theta, \text { given that } \cos \theta=\frac{2}{3}$$

Step-by-Step Solution

Answer

\( \sec \theta = \frac{3}{2} \)

Step 1: Recall the Reciprocal Identity for Secant

The secant of an angle \( \theta \) is the reciprocal of the cosine of that angle. We can use the identity:\[ \sec \theta = \frac{1}{\cos \theta} \] This identity allows us to find \( \sec \theta \) if we know \( \cos \theta \).

Step 2: Substitute the Given Value of Cosine

We are given that \( \cos \theta = \frac{2}{3} \). We substitute this value into our reciprocal identity:\[ \sec \theta = \frac{1}{\frac{2}{3}} \]

Step 3: Simplify the Expression

When simplifying \( \frac{1}{\frac{2}{3}} \), we multiply by the reciprocal of the denominator:\[ \sec \theta = 1 \times \frac{3}{2} = \frac{3}{2} \]

Step 4: Check for Rationalization

Since the expression \( \frac{3}{2} \) is already rational, there is no need to further rationalize the denominator or simplify the expression.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is essential in trigonometry. It's defined as the reciprocal of the cosine function. Understanding this concept simplifies finding unknown values in trigonometric problems when given other function values.
To clarify, if you know the cosine value of an angle, you can effortlessly find the secant by taking:

• The reciprocal of the cosine value.

This means, mathematically, that \( \sec \theta = \frac{1}{\cos \theta} \).
It is crucial to be comfortable with this identity, as it appears frequently, even in various calculus and physics problems. Hence, always remember that secant helps to "undo" the work of cosine, meaning it gives you a way to work with ratios of triangles or waves by flipping them over!

Cosine Function

The cosine function, represented as \( \cos \theta \), plays a fundamental role in trigonometry and is a core part of circular and triangle-based calculations. It's defined, in the context of a right triangle, as the ratio of the length of the adjacent side to the hypotenuse:

• Formula: \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).

In the realm of the unit circle, \( \cos \theta \) represents the x-coordinate of a point corresponding to an angle \( \theta \).
Given that \( \cos \theta \) is used to derive \( \sec \theta \), understanding how it functions is essential. This function helps determine the size of angles or distance in a wave, light, or sound problem, making it crucial for physics as well.
When using wave functions like \( \cos \theta \), you're looking at a part of a broader family of sine and tangent functions, which together form a comprehensive trigonometric toolkit.

Rationalizing Denominators

Rationalizing denominators is a technique used primarily to simplify expressions in order to make calculations easier, especially when dealing with radicals. However, in our exercise, since the answer \( \frac{3}{2} \) is already rational, no further steps are necessary here.
When confronting irrational denominators (like roots), rationalization involves converting them to rational numbers by multiplying the numerator and the denominator by a suitable factor. Let's consider an example:

• Suppose you have \( \frac{1}{\sqrt{3}} \), you can rationalize by multiplying numerator and denominator by \( \sqrt{3} \); thus, you get \( \frac{\sqrt{3}}{3} \).

This process ensures the expression is more manageable and easier to work with, especially in precise calculations. Even though in our given problem the result does not involve such a case, understanding this method prepares you for handling more complex equations with irrational numbers.