Problem 57 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 51-60: Show that the equation is not an identity. (Hint: Find one number for which the equation is false.) $$ \cos (-t)=-\cos t $$

Step-by-Step Solution

Answer

The equation \( \cos(-t) = -\cos(t) \) is not an identity as shown by \( t = 0 \), where it results in \( 1 \neq -1 \).

Step 1: Understand the Trigonometric Identity

Recall the trigonometric identity for cosine: \( \cos(-t) = \cos(t) \). This tells us that cosine is an even function, which means that the cosine of a negative angle is the same as the cosine of the positive angle.

Step 2: Evaluate the Given Expression

The given expression is \( \cos(-t) = -\cos(t) \). Based on the identity from Step 1, we know that this equation does not hold for all values of \( t \) because \( \cos(-t) \) should be equal to \( \cos(t) \), not \(-\cos(t) \).

Step 3: Select a Test Value for t

We need to choose a value of \( t \) for which we can show that \( \cos(-t) eq -\cos(t) \). A simple choice is \( t = 0 \) because it often simplifies trigonometric functions.

Step 4: Substitute the Test Value into the Equation

Substituting \( t = 0 \) into the left side of the equation, we have \( \cos(-0) = \cos(0) \). Since \( \cos(0) = 1 \), the left side evaluates to 1. On the right side, we have \(-\cos(0) = -1 \). Therefore, \( 1 eq -1 \), showing that the equation \( \cos(-t) = -\cos(t) \) is false for \( t = 0 \).

Step 5: Conclude the Verification

Since \( t = 0 \) leads to a false statement in the equation \( \cos(-t) = -\cos(t) \), the equation is not an identity, as it does not hold true for all values of \( t \).

Key Concepts

Even Functions

An even function is a type of mathematical function that maintains symmetry around the y-axis. This means if you were to fold the graph of the function along the y-axis, it would match on both sides. Even functions follow a specific rule where the function evaluated at negative x is equal to the function evaluated at positive x, represented by the formula:

• For any even function: \[ f(-x) = f(x) \]

This property is crucial when working with trigonometric identities, as some trigonometric functions, including cosine, illustrate this even property precisely. Whenever you come across an equation or function, checking for symmetry can be a good first step to identify if it showcases evenness. This symmetrical property helps in simplifying expressions and solving equations effectively in trigonometry. Understanding this concept underpins much of the work you'll encounter with even functions such as cosine.

Cosine Function

The cosine function, one of the fundamental functions in trigonometry, is primarily known for its property of dealing with angles in standard position in the coordinate system. It is derived from the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
One of the distinguishing qualities of the cosine function is that it is an even function. This characteristic is encapsulated in the identity:

• \( \cos(-t) = \cos(t) \)

This confirms that the cosine of a negative angle is equivalent to the cosine of the positive angle. This property is not only crucial for understanding the behavior of the cosine function across different quadrants but also plays a significant role in verifying trigonometric identities. Cosine's periodic and repetitive nature influences many areas of mathematics and physics, especially in topics such as wave motion and harmonic oscillation. When dealing with trigonometric equations or expressions, recognizing these intrinsic properties of the cosine function can streamline calculations and verifications.

Verification of Equations

Verification of equations assists in determining whether a given equation is always valid (i.e., an identity) or not. To verify, one selects specific values to substitute into the equation, evaluating if the equation holds true in those instances. When checking trigonometric equations, like the one in the exercise \( \cos(-t) = -\cos(t) \), this process becomes particularly important.
By substituting a simple value, like \( t = 0 \), it's possible to show if the equation remains consistent or false with its expected trigonometric identity. For instance, substituting \( t = 0 \) yields \( \cos(-0) = \cos(0) \), simplifying to 1 on the left side, while the right evaluates to \(-1\), indicating the equation is false. This method reveals that the equation in question is not an identity because it doesn't hold for all values of \( t \). Verification is a key step in assessing trigonometric and algebraic equations, ensuring accuracy and understanding of mathematical properties and functions.