Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. $$\cos \theta, \text { given that } \sec \theta=-\frac{11}{7}$$

Step-by-Step Solution

Answer

\(\cos \theta = -\frac{7}{11}\).

Step 1: Understand Reciprocal Identities

The reciprocal identity for cosine and secant is \( \cos \theta = \frac{1}{\sec \theta} \). This identity will allow us to find \( \cos \theta \) from \( \sec \theta \).

Step 2: Substitute the Given Secant Value

We know \( \sec \theta = -\frac{11}{7} \). Using the reciprocal identity, substitute into \( \cos \theta = \frac{1}{\sec \theta} \), giving us \( \cos \theta = \frac{1}{-\frac{11}{7}} \).

Step 3: Simplify the Expression

To simplify \( \cos \theta = \frac{1}{-\frac{11}{7}} \), take the reciprocal of the secant, leading to \( \cos \theta = -\frac{7}{11} \).

Step 4: Finalize the Answer

Having simplified the expression correctly, we find that \( \cos \theta = -\frac{7}{11} \), meaning the reciprocal of \(-\frac{11}{7}\) simplifies to \(-\frac{7}{11}\).

Key Concepts

Cosine Function

The cosine function, often denoted as \( \cos \theta \), is one of the fundamental trigonometric functions. It relates an angle in a right triangle to the ratio of the length of the adjacent side to the hypotenuse. In the unit circle, that is a circle with radius 1 centered at the origin of a coordinate system, the cosine of an angle corresponds to the x-coordinate of a point on the circle itself.
\( \cos \theta \) helps solve many trigonometric problems and is particularly useful in physics, engineering, and computer graphics, thanks to its periodic nature, which repeats every \( 2\pi \) radians or 360 degrees.
When working with reciprocal identities in trigonometry, knowing the value of the cosine function becomes crucial, as it allows for seamless calculations and conversions among other trigonometric functions like secant, sine, and cosecant.

Secant Function

The secant function, represented as \( \sec \theta \), is another key trigonometric function. It is defined as the reciprocal of the cosine function:
\( \sec \theta = \frac{1}{\cos \theta} \).
This means that if you know the value of \( \cos \theta \), you can easily find \( \sec \theta \) by taking the reciprocal. This relationship is handy because it allows for straightforward trigonometric simplification and is particularly useful when dealing with angles and their properties.
The secant function is less commonly used compared to sine and cosine but plays an important role when dealing with reciprocal identities and trigonometric equations. It is particularly useful in simplifying expressions and solving trigonometric equations, particularly when simplifying complex fractions involving trigonometric functions.

Trigonometric Simplification

Trigonometric simplification involves using fundamental identities and relationships of trigonometric functions to simplify complex expressions. The use of reciprocal identities, such as \( \cos \theta = \frac{1}{\sec \theta} \), is a tool to achieve such simplification.
In the given exercise, we use the reciprocal identity to convert a known secant value into its corresponding cosine value, enabling further simplification into a more straightforward form. This often involves rationalizing denominators or expressing the function values in their simplest forms.
Simplifying trigonometric expressions is essential in mathematics to make equations easier to solve and understand, paving the way for clearer problem-solving steps and allowing us to gain insights into the properties and behaviors of mathematical models.