Problem 83 - A Graphical Approach to Precalculus with Limits

Question

If \(n\) is an integer, \(n \cdot 180^{\circ}\) represents an integer multiple of \(180^{\circ},(2 n+1) \cdot 90^{\circ}\) represents an odd integer multiple of \(90^{\circ}\), and so on. Decide whether each expression is equal to \(0,1\), or \(-1\) or is undefined. $$\cos \left[(2 n+1) \cdot 180^{\circ}\right]$$

Step-by-Step Solution

Answer

The expression equals \(-1\).

Step 1: Understanding the Expression

We need to evaluate \( \cos \left[(2n+1) \cdot 180^{\circ}\right] \). The expression involves the cosine of an angle that is an odd multiple of \(180^{\circ}\).

Step 2: Properties of Cosine Function at Multiples of 180°

The cosine function has specific values at integer multiples of \(180^{\circ}\). Specifically, for any integer \(k\), \(\cos(k \cdot 180^{\circ}) = \cos(180^{\circ} \cdot (2m+1)) = -1\).

Step 3: Replacing n with its Expression

For \((2n+1)\), this will always result in an odd integer when \(n\) is an integer. The expression can be written as an odd integer \(m\) such that \(m = 2n+1\).

Step 4: Evaluate the Cosine Function

Now the expression becomes \(\cos(m \cdot 180^{\circ})\) where \(m\) is odd. Thus, \(\cos((2n+1) \cdot 180^{\circ}) = \cos(180^{\circ})\), which simplifies to \(-1\).

Step 5: Final Result

Therefore, \(\cos \left[(2n+1) \cdot 180^{\circ}\right] = -1\) for any integer \(n\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is used to relate the angles of a right triangle to its side lengths, but its applications extend far beyond right triangles.
The cosine function is periodic, meaning it repeats its values in a regular interval. Specifically, the cosine function has a period of \(360^{\circ}\) or \(2\pi\) radians. This characteristic is vital when dealing with angles that are beyond the typical \(0^{\circ} \, \text{to} \, 360^{\circ}\) range.
Cosine values range from -1 to 1. For example:

• \(\cos(0^{\circ}) = 1\)
• \(\cos(90^{\circ}) = 0\)
• \(\cos(180^{\circ}) = -1\)
• \(\cos(270^{\circ}) = 0\)
• \(\cos(360^{\circ}) = 1\)

This cyclical nature is why the cosine function can be conveniently applied to multiples and integer multiples of angles.

Multiples of Angles

Understanding multiples of angles is essential when evaluating trigonometric functions.
A multiple of an angle means that the angle is multiplied by some constant factor. In the given exercise, these multiples may manifest as integer, half, or any fractional multiples.
For the cosine function, multiples of \(180^{\circ}\) are particularly significant because they often appear in standard angle evaluations. When evaluating \(\cos(k \cdot 180^{\circ})\) for any integer \(k\), the result depends on the multiple whether it's odd or even:

• For even multiples (e.g., \(0^{\circ}, 360^{\circ}\)), \(\cos\) yields \(1\).
• For odd multiples (e.g., \(180^{\circ}, 540^{\circ}\)), \(\cos\) yields \(-1\).

This relationship is a crucial piece of knowledge to quickly determine the result of the cosine function's value when dealing with multiples of \(180^{\circ}\).

Integer Multiples

Integer multiples play a pivotal role in simplifying trigonometric expressions. An integer multiple can be understood as multiplying a number by an integer where the result expresses a repeated addition of the base angle.
In our exercise, expressions like \(n \cdot 180^{\circ}\) indicate any multiple of \(180^{\circ}\) when \(n\) is an integer. This simplifies the evaluation of trigonometric functions as these multiples align with key angles on the unit circle.
Particularly, note how odd integer multiples such as \(m = 2n + 1\) produce results that culminate when multiplied with \(180^{\circ}\), leading to different but predictable cos values:

• The odd stature of the factor (e.g., \(\cos(180^{\circ})\)) signifies it will yield \(-1\).

Understanding these integer multiples allows for the easy evaluation of trigonometric expressions by leveraging the periodic properties of functions like cosine.