Problem 57 - Precalculus: Functions and Graphs

Question

Show that the equation is not an Identity. $$\cos (-t)=-\cos t$$

Step-by-Step Solution

Answer

The equation is not an identity; for example, it fails when \( t = 0 \).

Step 1: Understand the Identity

An identity is a mathematical equation that holds true for all possible values of the variable. To prove a given equation is not an identity, we must find at least one instance where it doesn't hold true.

Step 2: Review Trigonometric Properties

Recall the trigonometric property: \( \cos(-t) = \cos(t) \). This property suggests that the cosine function is an even function, meaning it is symmetric with respect to the y-axis.

Step 3: Compare the Given Equation

The equation given is \( \cos(-t) = -\cos(t) \). Compare it against the known property \( \cos(-t) = \cos(t) \). It suggests something contradictory, as instead of equality, it suggests \( \cos(-t) \) is the negative of \( \cos(t) \).

Step 4: Substitute a Specific Value

To determine if the equation is not an identity, try substituting a specific value for \( t \). Choose \( t = 0 \) because it is simple and often highlights differences. Substituting gives \( \cos(-0) = -\cos(0) \), which simplifies to \( 1 = -1 \), a false statement.

Step 5: Conclusion

Since substituting \( t = 0 \) gives a false statement, \( \cos(-t) = -\cos(t) \) is not true for all values of \( t \). Thus, the equation \( \cos(-t) = -\cos(t) \) is not an identity.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that is often denoted as \( \cos \theta \), where \( \theta \) represents an angle. It is part of the basic trio of trigonometric functions which include sine, cosine, and tangent. In the context of a right triangle, it represents the ratio of the length of the adjacent side to the hypotenuse. This function can be visualized as the horizontal coordinate of a point on the unit circle, which is particularly useful in defining cosine for any angle, not just those in right triangles.

Key features of the cosine function include:

• It is periodic with a period of \( 2\pi \). This means that the function repeats its values every \( 2\pi \) radians.
• The amplitude, or the maximum value, of the cosine function is 1. The minimum value is -1.

The function is widely used in various fields, including engineering, physics, and mathematics, because of its ability to model cyclic phenomena such as waves and oscillations.

Trigonometric Properties

Trigonometric properties are rules and formulas that define the relationships between the trigonometric functions such as sine, cosine, and tangent. These properties help in simplifying expressions, solving equations, and understanding the nature of these functions.

Some essential trigonometric properties include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \), which follows from the definition of sine and cosine in the unit circle.
• Reciprocal Identities: Relate the basic trigonometric functions to their reciprocals, such as secant being the reciprocal of cosine.
• Even-Odd Identities: These are particularly relevant in our context. They describe the behavior of trigonometric functions when their angle arguments are negated. For cosine, \( \cos(-\theta) = \cos(\theta) \), confirming its even nature.

Understanding and applying these properties enable students to manipulate and analyze trigonometric expressions efficiently, aiding in the solution of complex problems.

Even and Odd Functions

Mathematically, functions can be categorized as even, odd, or neither. This classification is based on their symmetry properties:

• Even Functions: These are symmetric about the y-axis, meaning \( f(-x) = f(x) \). The cosine function is an example of an even function.
• Odd Functions: These exhibit rotational symmetry about the origin, with the property \( f(-x) = -f(x) \). The sine function is a classic example of an odd function.

The significance of knowing whether a function is even or odd lies in simplifying expressions and solving equations. For instance, understanding that \( \cos(-t) = \cos(t) \) helps us quickly identify that the equation \( \cos(-t) = -\cos(t) \) is not valid for all \( t \), hence not an identity.

Moreover, by leveraging these properties, one can better interpret graphs, integrate functions over symmetric intervals, and solve real-world problems involving symmetrical behaviors.