Question

Give the intervals on which the given function is continuous. $$ h(t)=\cos t $$

Short answer

The function is continuous on \((-\infty, \infty)\).

Step-by-step solution

Understand the function

The function given is \( h(t) = \cos t \), which is a trigonometric function. Cosine is one of the basic trigonometric functions, and it is defined for all real numbers.

Identify continuity of cosine function

The cosine function, \( \cos t \), is continuous everywhere because it is defined for all real numbers without any breaks, jumps, or holes. In mathematical terms, it can be evaluated at every point on the real number line, and there are no discontinuities.

Determine the intervals of continuity

Since \( \cos t \) is continuous for every real number \( t \), the interval on which the function is continuous is \( (-\infty, \infty) \).

Key concepts

Trigonometric Functions

Trigonometric functions are a fundamental part of mathematics, especially in the study of geometry and periodic phenomena. These functions relate the angles of a triangle to the lengths of its sides. The most common trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each of these functions takes an angle as input and provides a specific ratio derived from the sides of a right triangle as output.

• Sine Function: Represents the ratio of the length of the side opposed to the angle to the hypotenuse.
• Cosine Function: Represents the ratio of the length of the adjacent side to the angle to the hypotenuse.
• Tangent Function: Represents the ratio of the sine of the angle to the cosine of the angle.

These functions are periodic, meaning they repeat their values in a regular pattern. Understanding trigonometric functions is crucial, as they are widely used in physics, engineering, and other fields involving wave-like phenomena.

Cosine Function

The cosine function, denoted as \( \cos t \), is a trigonometric function that describes a specific relationship within right-angled triangles. Cosine specifically measures the horizontal component of an angle, when the hypotenuse (the longest side) is known. This function takes any real number as input, corresponding to an angle measure, and outputs a value between \(-1\) and \(1\).

Cosine is particularly known for its smooth, wave-like graph, which is periodic with a period of \(2\pi\). This periodicity means every \(2\pi\) radians (or 360 degrees), the function repeats its values.

• The graph of \( \cos t \) starts at \(1\) when \(t = 0\), dips to \(-1\) at \(t = \pi\), and returns to \(1\) at \(t = 2\pi\)
• It is symmetric about the vertical axis, demonstrating its even nature (\(\cos(-t) = \cos(t)\)).

This symmetric property is fundamental in various applications such as signal processing and in calculating Fourier transforms.

Continuous Everywhere

A function being continuous everywhere means that there are no breaks, jumps, or holes in its graph across its domain. The cosine function, \( \cos t \), exemplifies this property magnificently. It is defined at every point along the real number line, making it continuous at all real numbers.

• Definition: A function is continuous at a point \(c\) if the limit of the function as it approaches \(c\) equals the function's value at \(c\).
• The cosine function exceeds this definition, being smooth and uninterrupted everywhere from \(-\infty\) to \(\infty\).

This attribute of continuity is crucial not only for ensuring mathematical rigor but also in practical applications such as computer graphics and engineering. Continuous functions enable predictable and stable modeling of real-world phenomena, which is invaluable in precise calculations and simulations.