Question

\(g(x)=\csc \frac{\pi}{4} x\)

Short answer

The function \(g(x)= \csc\left(\frac{\pi}{4} x\right)\) is defined for all \(x\) except \(x=4n\) where \(n\) is an integer. The value of \(g(x)\) is the reciprocal of the sine of \(\frac{\pi}{4} x\).

Step-by-step solution

Review the definition of a cosecant

The cosecant \(\csc(x)\) is defined as the reciprocal of the sine function, i.e., \(\csc(x) = 1/\sin(x)\). The value of \(\csc(x)\) is undefined when \(sin(x) = 0\).

Determine the undefined points

The sine function is zero when its argument is a multiple of \(\pi\). Therefore, the value of \(x\) should not make the argument \(\frac{\pi}{4} x\) equal to \(n\pi\) where \(n\) is an integer. For all other values of \(x\), \(g(x)\) will be defined and its value will be the reciprocal of the sine of \(\frac{\pi}{4} x\). So the equation \(\frac{\pi}{4} x = n\pi\) yields \(x = 4n\), for \(n\) as an integer.

Determining the function

Since there is no other condition given in the problem, every value of \(x\) except \(x=4n\) where \(n\) is an integer, will satisfy the function \(g(x)\). The actual value of \(g(x)\) for each \(x\) would be \(g(x)=\csc \frac{\pi}{4} x = \frac{1}{\sin(\frac{\pi}{4} x)}\).

Key concepts

Cosecant Function

The cosecant function is an interesting and important part of trigonometry. It is not as commonly discussed as the sine or cosine functions, but it serves a very specific purpose in mathematics. The cosecant of an angle, denoted as \( \csc(x) \), is defined as the reciprocal of the sine function. This means:

• \( \csc(x) = \frac{1}{\sin(x)} \)

Relating to the unit circle, for any angle \( x \), the cosecant function provides the hypotenuse length in relation to the opposite side of the angle within a right triangle. It's important to note its characteristics:

• Cosecant is undefined wherever the sine of the angle is zero.
• It has the same sign as the sine, meaning it is positive when sine is positive, and negative when sine is negative.

Sine Function

The sine function is one of the essential trigonometric functions, commonly used across mathematics. It is represented as \( \sin(x) \) for an angle \( x \), and can be visualized on the unit circle where it indicates the y-coordinate. In a right-angled triangle, it represents the ratio of the length of the side opposite the angle over the hypotenuse.

• The function has a range between \(-1\) and \(1\).
• It is periodic with a period of \(2\pi\).
• The sine of an angle is zero at integral multiples of \(\pi\), i.e., \( \sin(n\pi) = 0 \).

These properties make the sine function quite versatile and useful in various applications, including solving physics problems and modeling periodic phenomena.

Undefined Points

Undefined points occur in mathematics when a function does not have a meaningful value at certain inputs. In the case of the cosecant function, these points occur when the sine function results in zero, as dividing by zero is undefined.

• For \( \csc(x) = \frac{1}{\sin(x)} \), if \( \sin(x) = 0 \), the expression becomes undefined.
• The values of \( x \) that make \( \sin(x) = 0 \) are integers of \( \pi \), such as \( 0, \pi, 2\pi, \ldots \).

Understanding undefined points is crucial. It helps in determining where certain functions do not make sense or need careful consideration, especially in other disciplines that use mathematics, like engineering and computer science.

Reciprocal Function

In mathematics, the reciprocal function transforms a number into its multiplicative inverse. For a given number \( a \), its reciprocal is \( \frac{1}{a} \). When you apply this concept to functions, you transform the value of a function into its reciprocal.

• The reciprocal function \( f(x) = \frac{1}{g(x)} \) is not defined wherever \( g(x) = 0 \).
• This principle is applied in trigonometry, where the cosecant is the reciprocal of the sine function: \( \csc(x) = \frac{1}{\sin(x)} \).
• Understanding reciprocals is an algebraic skill that supports solving equations and understanding function behavior.

Reciprocal functions provide great insights into behavior changes in symmetry and asymptotes, especially in graphing and calculus.