Question

What is the range of the cosine function?

Short answer

The range of the cosine function is [-1, 1].

Step-by-step solution

Understand the Cosine Function

The cosine function, denoted as \(\text{cos}(x)\), is a periodic function that relates the angle \(\theta\) in a right-angled triangle to the ratio of the length of the adjacent side to the hypotenuse.

Determine the Behavior of Cosine

The cosine function oscillates between -1 and 1. This means that for any value of \(\theta\), \(\text{cos}(\theta)\) will always fall within this range.

Identify the Range

Since the highest value \(\text{cos}(x)\) can achieve is 1 and the lowest value is -1, the range of the cosine function is between -1 and 1.

Key concepts

cosine function

The cosine function is one of the fundamental trigonometric functions in mathematics. It is denoted as \(\text{cos}(x)\) and plays an important role in various fields, including geometry, physics, and engineering.

The cosine function relates the angle in a right-angled triangle to the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. This can be visualized easily when you think about the unit circle, where the x-coordinate of a point on the circle corresponds to the cosine of the angle formed.

For example, if you have an angle \( \theta \), the cosine function allows you to find the x-coordinate of the point on the unit circle. This makes the cosine function extremely useful in solving geometric problems and modeling periodic phenomena in the real world.

range of a function

The range of a function refers to the set of all possible output values (y-values) that the function can produce.
It is crucial to understand the range, as it helps in determining the limitations and behavior of the function.

For the cosine function, we can observe that it oscillates between -1 and 1. This is because the cosine function is bounded, meaning it cannot produce any values outside this interval. Thus, the range of the cosine function is \( -1 \leq \cos(x) \leq 1 \).

Knowing the range is essential in understanding the function's capabilities and constraints, especially when solving equations or analyzing real-world scenarios that involve the cosine function.

periodic functions

Periodic functions are functions that repeat their values in regular intervals or periods. The cosine function is a perfect example of a periodic function.

A function \( f(x) \) is considered periodic if there exists a positive constant \( T \) such that \( f(x + T) = f(x) \) for all \( x \). For the cosine function, this period is \( 2\text{π} \), meaning \( \text{cos}(x + 2\text{π}) = \text{cos}(x) \).

This periodic nature makes the cosine function extremely useful in modeling cyclic phenomena such as waves, oscillations, and alternating currents.

Understanding that the cosine function is periodic helps to predict its behavior over repeated intervals, facilitating easier solving of problems related to cycles and periodic processes.