Problem 84 - 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. $$\cos \left(n \cdot 360^{\circ}\right)$$

Step-by-Step Solution

Answer

The expression is equal to 1.

Step 1: Understand the Problem

We need to determine if the expression \( \cos(n \cdot 360^{\circ}) \) is equal to 0, 1, -1, or undefined. This expression involves cosine of an integer multiple of \(360^{\circ}\).

Step 2: Cosine Property

The cosine function is periodic with a period of \(360^{\circ}\). This means that for any integer \(n\), \(\cos(n \cdot 360^{\circ}) = \cos(0^{\circ})\).

Step 3: Evaluate Cosine at Zero Degrees

The cosine of \(0^{\circ}\) is \(1\). Hence, \(\cos(n \cdot 360^{\circ}) = \cos(0^{\circ}) = 1\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions used to relate the angle of a triangle to the ratios of two of its sides. Specifically, in a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. This function is broadly used in various fields, including physics, engineering, and computer graphics.
Key aspects of the cosine function include:

• It is an even function, meaning that \(\cos(-x) = \cos(x)\).
• The cosine function takes on values between -1 and 1.
• It is highly associated with the unit circle, where any angle \(\theta\) can be placed in the circle, and the x-coordinate of the corresponding point on the circle is the cosine of the angle.

Understanding these properties helps in solving trigonometric problems, such as determining the value of \(\cos(n \cdot 360^\circ)\). Recognizing that the cosine function's output will repeat every full rotation illustrates how we can find similar cosine values even as the angles become very large.

Periodic Functions

Periodic functions, like the cosine function, repeat their values in regular intervals. These functions are significant in many real-world applications, such as signal processing and wave motion. The cosine function is periodic with a period of \(360^\circ\) or \(2\pi\) when using radians.
Key features of periodic functions include:

• The basic repeating interval is known as the 'period'.
• For the cosine function, this means \(\cos(\theta + 360^\circ) = \cos(\theta)\).
• These functions are continuous, meaning they create smooth and uninterrupted graphs.

The periodic nature of cosine ensures that transformations like stretching or translating the graph do not distort its fundamental pattern. In our exercise problem, \(\cos(n \cdot 360^\circ)\) benefits from this property since it simplifies to \(\cos(0^\circ)\), which equals 1, regardless of how large \(n\) is as long as it remains an integer.

Angle Measurement

Angle measurement is crucial in understanding trigonometric functions and their applications. Angles can be measured in degrees or radians, with each measure offering different benefits based on the problem context.
Some essential points about angle measurement include:

• Degrees are often more intuitive, where a full circle is \(360^\circ\).
• Radians are more mathematically natural for calculus, where a full circle is \(2\pi\) radians.
• Converting between these units is straightforward: \(180^\circ = \pi \, \text{radians}\).

In trigonometry, angles are frequently used to determine positions on the unit circle, helping find values for cosine and other trigonometric functions. In our exercise, the angle \(n \cdot 360^\circ\) shows how multiples of a full rotation return us to the beginning position, making it easier to calculate simple cosine values.