Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\cos (-5.46)$$

Step-by-Step Solution

Answer

\( \cos(-5.46) = \cos(5.46) \)

Step 1: Identify the Trigonometric Identity

Recognize that cosine is an even function. The property of even functions is that \[ \cos(-x) = \cos(x) \]. We'll use this identity to rewrite the given expression.

Step 2: Apply the Even Identity

Apply the even identity to the given expression: \[ \cos(-5.46) = \cos(5.46) \]. This shows that the cosine of a negative angle is the same as the cosine of the positive angle.

Step 3: Verify the Positive Argument

The expression has been rewritten as a function of a positive angle: \[ \cos(5.46) \] is the cosine of the positive angle 5.46. This satisfies the requirement of the problem.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are mathematical functions that play a crucial role in various areas like geometry, calculus, and physics. They relate angles of a triangle to the lengths of its sides and are periodic, meaning they repeat their values in regular intervals. The most common trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Another important feature is that they have specific properties, like evenness and oddness, which help in simplifying expressions and solving problems.
These functions are crucial for understanding wave patterns, circular motion, and oscillations. Understanding the behavior of trigonometric functions, like their periodicity and symmetry, helps to analyze angles and functions efficiently.

• Sinusoidal patterns observed in nature and engineering are often modeled using trigonometric functions.
• They provide a means to handle angular calculations precisely and effectively.

Knowing how they operate and transform, such as the even and odd identities, opens the door to solve complex problems in a straightforward manner.

Properties of Even Functions

An even function is a type of mathematical function where the function values are symmetric about the y-axis. This means that for any input value \( x \), the function satisfies the condition \( f(-x) = f(x) \).
Cosine is one of the primary examples of an even function. Since \( \cos(-x) = \cos(x) \), this property can be extremely useful when simplifying expressions involving negative angles. Using these rules allows for rewriting expressions more conveniently.
Even functions reflect symmetry across the vertical line at \( x = 0 \), meaning that their graph remains unchanged if flipped horizontally:

• The graph's shape remains constant, indicating symmetry.
• It offers a straightforward approach to simplify calculations involving negative values.

Recognizing that the cosine function holds this symmetry can help in evaluating expressions and rewriting them using positive angles.

Working with Negative Angles in Trigonometry

Negative angles can often seem confusing, but in trigonometry, they become manageable with the use of identities. When dealing with trigonometric functions, negative angles typically imply rotation in the opposite direction of positive angles. For instance, when considering angles in a circle, negative angles move clockwise, whereas positive angles move counterclockwise.
Understanding the properties of trigonometric functions, like even and odd identities, helps to handle these cases effectively. With even functions like cosine, replacing a negative angle with a positive one does not change the function's value:- The expression \( \cos(-5.46) \) becomes \( \cos(5.46) \), simplifying it to incorporate only positive values.- The transformation relies on the symmetry of these functions, reducing complexity.
By understanding these properties, we simplify our calculations significantly, transforming negative angles into a simpler form to work through trigonometric equations more easily.