91Ó°ÊÓ

The cosecant function, denoted as \(\csc x\), is one of the fundamental trigonometric functions and is defined as the reciprocal of the sine function. Mathematically, it is expressed as:\[ \csc x = \frac{1}{\sin x} \]This means that wherever the sine of an angle is zero, the cosecant function is undefined. This happens at specific points such as \(x = 0, \pi, 2\pi\), and so on.

• **Range and Domain:** The range of \(\csc x\) is the set of all real numbers except for the open interval \((-1, 1)\). This is because the sine function has values that range from -1 to 1, except that \(\csc x\) is not defined at these values.
• **Key Angles:** Often, when solving trigonometric equations involving cosecant, we look for angles where sine equals commonly known values like \(-\frac{1}{2}\), or \(\frac{\sqrt{2}}{2}\).

Understanding the cosecant function is critical in solving equations like the original exercise, where we had to find solutions for expressions involving \(\csc x\). By knowing the relationship between sine and cosecant, we can easily manipulate and solve such equations.
For example, if \(\csc x = -2\), this implies that \(\sin x = -\frac{1}{2}\). This reciprocal nature aids us in finding solutions over given intervals.

The zero-product property is a fundamental algebraic principle stating that if the product of two or more factors is equal to zero, then at least one of the factors must be zero. This property underpins the solving of many algebraic equations and is equally applicable to trigonometric equations.
In our exercise, we used the zero-product property to solve:\[ (\csc x + 2)(\csc x - \sqrt{2}) = 0 \]By the zero-product property:

• Set \(\csc x + 2 = 0\)
• Set \(\csc x - \sqrt{2} = 0\)

Each equation can be solved independently to find the possible values of \(x\). The solutions to these individual equations comprise the complete solution to the original equation.
This approach is extremely useful because it simplifies complex equations into manageable parts, allowing us to find potential solutions more quickly.

Interval notation is a method of denoting a range of numbers, useful in mathematics to describe solutions or values belonging to a particular set. In the context of trigonometric equations, interval notation helps specify over which values we are interested in finding solutions, such as \[ [0, 2\pi) \]Here's how to understand this interval:

• The "[" symbol means that the starting point, 0, is included in the interval.
• The ")" symbol means that the endpoint, \(2\pi\), is not included, indicating an open interval at \(2\pi\).

This often defines the domain over which we are examining our solutions. In trigonometry, solving within a specific interval is crucial as trigonometric functions are periodic and their solutions repeat over different intervals. Thus, specifying a particular interval, such as \([0, 2\pi)\), prevents contradiction with overlapping solutions from other periods.
Comprehending interval notation makes problem-solving in mathematics efficient and clear, ensuring we consider only the relevant solutions.