Problem 103 - Precalculus

Question

Determine whether each statement is true or false. \(\cos \theta=\cos \left(\theta+360^{\circ} n\right),\) where \(n\) is an integer.

Step-by-Step Solution

Answer

True, due to the periodic nature of the cosine function with period \(360^{\circ}\).

Step 1: Understanding the periodicity of cosine

The cosine function, \(\cos \theta\), is periodic with a period of \(360^{\circ}\). This means that its value repeats every \(360^{\circ}\). This property can be written as \(\cos \theta = \cos (\theta + 360^{\circ}k)\), where \(k\) is any integer.

Step 2: Applying the periodic property to the problem

In the given statement, we have \(\cos \theta = \cos (\theta + 360^{\circ}n)\). Since \(\cos(\theta + 360^{\circ}n)\) uses the periodicity property of the cosine function with \(n\) as an integer, the statement is consistent with the definition of periodicity.

Step 3: Conclusion about the statement

Since the statement \(\cos \theta = \cos (\theta + 360^{\circ}n)\) holds true by the periodic property of the cosine function, each instance where \(n\) is an integer will result in an equivalent expression.

Key Concepts

Understanding Periodicity

Periodicity is a fundamental concept in trigonometry that helps us understand how certain functions, like the cosine function, behave over their domain. When we say a function is periodic, it means the function repeats its values at regular intervals, known as periods. For the cosine function, the period is exactly 360 degrees.

To better grasp periodicity, imagine walking around a circle. After completing a full circle, you return to the starting point, having covered an angle of 360 degrees. Similarly, the cosine function repeats its values after every 360 degrees.

This periodic behavior is not just limited to degrees but can be applied to the radian measure as well, where the period would be computed as \(2\pi\) radians. This concept of periodicity ensures that the trigonometric functions remain predictable and calculable over infinite domains.

Exploring the Cosine Function

The cosine function, often denoted as \(\cos \theta\), is a critical part of trigonometry. Cosine is defined as the horizontal coordinate of a point on the unit circle at an angle \(\theta\) from the positive x-axis.

Important properties related to the cosine function include its even symmetry and its range of values. An even function means that \(\cos(\theta) = \cos(-\theta)\). As for its range, the cosine function outputs values between -1 and 1 for any real number input.

• The maximum value: 1, occurs when \(\theta\) is 0, 360, etc.
• The minimum value: -1, occurs when \(\theta\) is 180, 540, etc.

Such properties make the cosine function predictable and consistent, making it useful in solving equations and modeling periodic phenomena in real life.

Role of Integer Values in Periodicity

Integer values play a vital role when dealing with the periodicity of trigonometric functions. Consider the expression \(\theta + 360^{\circ}n\), where \(n\) is an integer. Here, the integer \(n\) determines how many full cycles of 360 degrees you "add" to \(\theta\).

If \(n=1\), you add one full circle (or period) to your angle, which does not change the cosine value due to periodicity. If \(n=0\), you remain at the initial angle without change, while negative values of \(n\) imply retracing the cycles backward.

This characteristic of using integer multiples is beneficial for creating trigonometric identities and simplifies calculations, especially when analyzing wave patterns or oscillations. Understanding how integer values interact with periodic functions aids in deeply understanding cycles in mathematics and nature.