91Ó°ÊÓ

Trigonometric functions are fundamental in mathematics, especially in studying periodic phenomena. One of the most commonly used trigonometric functions is the cosine function, often represented as \(\text{cos} \theta\). It expresses the relationship between the angle \(\theta\) and the x-coordinate of a point on a unit circle. The cosine function has several key properties that make it extremely useful.

For example:

• It is periodic, meaning it repeats its values at regular intervals.
• It is an even function, so \(\text{cos}(-\theta) = \text{cos}(\theta)\).

By understanding these properties, students can solve a variety of mathematical problems involving periodicity and symmetry.

The range of a function refers to all the possible values that the function can take. In the case of the cosine function, \(\text{cos} \theta\), its range is restricted between -1 and 1. This means that no matter what value \(\theta\) takes, \(\text{cos} \theta\) will always lie within this interval, formally expressed as \[-1 \leq \text{cos} \theta \leq 1\].

Understanding the range is crucial for various applications, including signal processing, physics, and engineering. It ensures you know the limits within which the function operates, preventing errors in calculations and interpretations.

In our given problem, since \(\text{cos}(\theta)\) always lies between -1 and 1, the function \( f(\theta) = \text{cos}(\theta) \) will also share this range. Simply put, the range of \(f(\theta) \), given \(f(\theta) = \text{cos}(\theta) \), is \[-1 \leq f(\theta) \leq 1\].

Interval notation is a way of expressing the set of all numbers between two endpoints. It is widely used in mathematics to describe ranges of functions and solutions to inequalities.

There are two main types of interval notations: closed intervals and open intervals. A closed interval, denoted \([a, b]\), includes its endpoints, meaning that \(a \leq x \leq b\). Conversely, an open interval, denoted \( (a, b) \), does not include its endpoints, meaning that \( a \< x \< b\).

In the context of the cosine function, the range is written using closed interval notation as \[-1, 1\] because \(\text{cos}(\theta)\) can indeed be -1, 0, or 1, inclusive.

Using interval notation simplifies the representation and understanding of function ranges and is a crucial tool for algebra and calculus students. It's especially useful for depicting solution sets succinctly.