Problem 132 - Precalculus

Question

For the following exercises, find the requested value. State the domain of the sine and cosine functions.

Step-by-Step Solution

Answer

The domain for sine and cosine functions is all real numbers: \\((-\infty, \infty)\\).

Step 1: Understanding the Functions

The sine and cosine functions are fundamental trigonometric functions often encountered in various mathematical problems. They are usually described over angles, often measured in radians or degrees, and output values between -1 and 1 inclusive.

Step 2: Identifying the Domain

The domain of a function includes all possible input values (or angles, in the case of trigonometric functions) for which the function is defined. For both the sine and cosine functions, the inputs are angle measures which can be any real number.

Step 3: Expressing the Domain Mathematically

Since sine and cosine functions can take any real number as input, we express this domain mathematically as all real numbers. In interval notation, this is represented as \((-\infty, \infty)\).

Key Concepts

sine function

The sine function, often denoted as \(\sin \theta\), is a trigonometric function that outputs values between -1 and 1 inclusive.
The sine function is periodic with a period of \(2\pi\), meaning that it repeats its values every \(2\pi\) radians.
It is essential for analyzing oscillations, waves, and circular motion.Key characteristics of the sine function include:

• Range: [−1, 1]
• Period: \(2\pi\)
• Odd function: Symmetric around the origin, satisfying \(\sin(-\theta) = -\sin(\theta)\)

Remember, the sine function applies to any angle, and its usefulness in mathematical modeling goes beyond theoretical calculations, representing real-world phenomena.

cosine function

The cosine function, represented as \(\cos \theta\), is another fundamental trigonometric function.
Similar to the sine function, it also results in values ranging from -1 to 1, and it is periodic with a period of \(2\pi\).
This function is invaluable in measuring angles and modeling periodic behavior in various fields.Characteristics of the cosine function include:

• Range: [−1, 1]
• Period: \(2\pi\)
• Even function: Satisfies \(\cos(-\theta) = \cos(\theta)\), symmetric with respect to the y-axis.

These properties make the cosine function important for understanding phenomena like alternating current in electrical engineering or sound waves in acoustics.

real numbers

Real numbers encompass the entire spectrum of numbers that you use in everyday mathematics and scientific computations. These include:

• Rational numbers: Such as 1/2, 5, and -3.
• Irrational numbers: Such as \(\sqrt{2}\) or \(\pi \).
• Whole numbers and integers.

Trigonometric functions like sine and cosine are defined for all real numbers. This means you can plug in any positive, negative, integer, or non-integer value into \(\sin\) and \(\cos\) functions, and expect a meaningful output.
Understanding real numbers is crucial, as they form the foundation upon which the continuity and periodicity of trigonometric functions are based.

interval notation

Interval notation is a mathematical form used to represent subsets of real numbers.
It is particularly useful in denoting domains and ranges of functions, allowing clear communication of which values are included in a set.Common symbols in interval notation include:

• '\((...)\)' for open intervals, meaning the end values are not included
• '\([...]\)' for closed intervals, encompassing the end values

For the sine and cosine functions, the domain is all real numbers, represented in interval notation as \((−\infty, \infty)\). This succinct representation signifies every number from negative infinity to positive infinity, highlighting the functions' continuous nature over all possible angles.