Question

\(g(x)=2 \csc x\)

Short answer

The given function \(g(x) = 2\csc x\) is the trigonometric cosecant function. It can be rewritten in terms of sine as \(g(x) = \frac{2}{\sin x}\). Values of x are evaluated by substituting them in this function.

Step-by-step solution

Understand the Cosecant Function

The cosecant function, represented as \(csc x\), is a trigonometric function. It is the reciprocal of the sine function, hence, it is written as \(\csc x = \frac{1}{\sin x}\). This means, the cosecant of an angle is the length of the hypotenuse divided by the length of the side opposite the angle in a right triangle.

Convert Cosecant to Sine Form

Since the given function is \(g(x) = 2\csc x\), using the reciprocal property of trigonometric functions, this function can be rewritten as \(g(x) = \frac{2}{\sin x}\). This step makes it easier to evaluate for values of \(x\).

Evaluate the Function

Once the function is in sine form, it can be used to evaluate for specific values of \(x\). For instance, to evaluate the function at \(x = \frac{\pi}{2}\), substitute \(\frac{\pi}{2}\) in place of \(x\) in the sine function to get \(g(\frac{\pi}{2}) = \frac{2}{\sin (\frac{\pi}{2})}\). The sine of \(\frac{\pi}{2}\) is 1, so the value of the function at this point is 2.

Key concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are fundamental in the study of periodic phenomena, such as waves and oscillations, as well as in the fields of engineering, physics, and geometry. The primary trigonometric functions are sine (\textt{sin}), cosine (\textt{cos}), and tangent (\textt{tan}), with each function providing a ratio between two sides of a right triangle.

Each trigonometric function corresponds to a specific ratio: \textt{sin} is the opposite side over the hypotenuse, \textt{cos} is the adjacent side over the hypotenuse, and \textt{tan} is the opposite side over the adjacent side. The functions of cotangent (\textt{cot}), secant (\textt{sec}), and cosecant (\textt{csc}), as seen in the exercise with the function \(g(x)=2 \csc x\), are the reciprocals of \textt{tan}, \textt{cos}, and \textt{sin}, respectively.

Reciprocal Identities

Reciprocal identities in trigonometry refer to the relationship between the primary trigonometric functions and their reciprocals. These identities are handy when simplifying complex trigonometric expressions or solving equations. One can think of a reciprocal identity as the 'flip' of a fraction.

The main reciprocal identities are:\[\cot \theta = \frac{1}{\tan \theta}\], \[\sec \theta = \frac{1}{\cos \theta}\], and \[\csc \theta = \frac{1}{\sin \theta}\]. In the context of the given exercise, \(g(x) = 2\csc x\) is re-expressed using the reciprocal identity of the sine function as \(g(x) = \frac{2}{\sin x}\). This transformation simplifies the process of function evaluation, as it is often easier to work with the well-known sine function values.

Function Evaluation

Function evaluation involves substituting a specific value into a function and calculating the output. It is a fundamental concept in algebra and calculus that allows for the understanding of how functions behave at different points. When evaluating trigonometric functions, this might involve using known values at special angles, like \(0\), \(\frac{\pi}{2}\), \(\pi\), and so forth.

In our case, to evaluate the function \(g(x) = 2\csc x\) at \(x = \frac{\pi}{2}\), we substitute \(x\) with \(\frac{\pi}{2}\) into the transformed function \(\frac{2}{\sin x}\) to find the value of \(g(\frac{\pi}{2})\). This evaluation heavily relies on the recognition of known values of \textt{sin}, particularly that \textt{sin}(\frac{\pi}{2}) = 1. Therefore, \(g(\frac{\pi}{2}) = \frac{2}{1} = 2\).

Sine Function

The sine function, usually expressed as \(\sin x\), is one of the primary trigonometric functions and is defined as the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right-angled triangle. It is also represented as a wave-like graph when plotted in a coordinate system, reflecting its regular periodicity. For angles between \(0\) and \(2\pi\), the \textt{sin} function takes values between \(-1\) and \(1\).

The sine function is periodic with a period of \(2\pi\), implying that \(\sin (x + 2\pi) = \sin x\text{ for all }x\). In the context of the original question, by understanding the sine function and its properties, we can manipulate the cosecant function with ease, such as inverting it to use traditional \textt{sin} values to evaluate \(g(x)\).