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Chapter 3: Problem 58

Identify the argument of each function. \(\cos \left(2 x-\frac{3 \pi}{2}\right)\)

Short Answer

Expert verified
The argument is \(2x - \frac{3\pi}{2}\).

Step by step solution

01

Analyze the Function

The given function is \[ \cos \left(2x - \frac{3\pi}{2} \right) \]We want to identify the argument of the cosine function. In trigonometry, the argument of a function is the expression inside the function.
02

Identify the Argument

For the cosine function above, the expression inside the function is:\[ 2x - \frac{3\pi}{2} \]This is the argument of the cosine function. It includes both the variable term \(2x\) and the constant term \(-\frac{3\pi}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function is one of the basic trigonometric functions. It is often abbreviated as "cos" and is used to determine the ratio of the length of the adjacent side over the hypotenuse in a right-angled triangle. The function can be applied to an angle, and it outputs a value between -1 and 1.

In mathematical terms, the cosine of an angle \( \theta \) is written as \( \cos(\theta) \). This function is periodic, meaning it repeats values in regular intervals. The most commonly studied interval is from 0 to \( 2\pi \).

  • The cosine function is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
  • The graph of the cosine function is a waveform and is symmetric about the y-axis.
  • Its values are crucial in various fields like physics, engineering, and computer science, besides just mathematics.
Understanding the properties and characteristics of the cosine function is crucial when exploring trigonometric expressions and solving related problems.
Trigonometric Expressions
Trigonometric expressions are mathematical phrases that involve trigonometric functions like sine, cosine, or tangent. These functions are applied to variables or angles, and the expressions can become quite complex, combining several trigonometric functions.

When dealing with an expression like \( \cos(2x - \frac{3\pi}{2}) \), we must identify its components.
  • "2x" is a term involving a variable and reveals the function's dependency on a changing angle.
  • "-\frac{3\pi}{2}" is a constant and represents a phase shift that affects the angle's starting point.
By carefully examining and breaking down these expressions, we can determine an angle's specific position on the trigonometric circle at any given time. This aids not only in solving mathematical problems, but also in understanding oscillatory systems and other real-world phenomena.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every possible value of the involved variables. These identities are crucial tools for simplifying complex trigonometric expressions and solving equations.

Some fundamental identities include:
  • Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
  • Angle Sum and Difference Identities: Particularly useful for finding the cosine or sine of sum or difference of two angles. For example, \( \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) \).
  • Double Angle Identity: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)
Understanding these identities allows us to manipulate and transform trigonometric expressions more easily. They are invaluable when proving equations, simplifying expressions, and solving trigonometric equations in more advanced mathematics.

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