Problem 111 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blanks. (Note: \(x \rightarrow c^{+}\) indicates that \(x\) approaches \(c\) from the right, and \(x \rightarrow c^{-}\) indicates that \(x\) approaches \(c\) from the left.) $$\text { As } x \rightarrow \frac{\pi}{2}, \sin x \rightarrow$$ ______ $$\text { and } \csc x \rightarrow$$ _____

Step-by-Step Solution

Answer

As \(x\) approaches \(\pi/2\), \( \sin x \) approaches 1 and \(\csc x \) also approaches 1.

Step 1: Analysing the Sine Function

As \(x\) approaches \(\pi/2\), we need to determine what the \( \sin x \) approaches. This involves understanding the basic graph of the sine function. Knowing that the sine function oscillates between -1 and 1, it reaches its maximum value at \(\pi/2\). Therefore, as \(x\) approaches \(\pi/2\) from either side, \( \sin x \) approaches 1.

Step 2: Analysing the Cosecant Function

The cosecant function, \(\csc x\), is the reciprocal of the sine function, \(\sin x\). This means as \(x\) approaches \(\pi/2\) from either side, \(\csc x \) approaches the reciprocal of 1, which is still 1.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It's often abbreviated as "sin" and is primarily used to relate angles to the corresponding ratios of sides in a right-angled triangle. The sine of an angle \(\theta\) in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. \[ \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \]It's important to note that the sine function oscillates between -1 and 1 for any angle \(x\), which reflects its periodic nature. This cycle repeats every \(2\pi\) radians, making it a cyclic graph. When looking at the unit circle, the sine value represents the \y\-coordinate at a given angle.The point \(x = \frac{\pi}{2}\) is especially crucial because here the sine function reaches its peak value of 1. This characteristic is pivotal in evaluating limits involving sine, as shown in the problem where \(\sin x\) approaches 1 as \(x\) gets closer to \(\frac{\pi}{2}\). Understanding this behavior can further help in determining the behavior of related functions like cosecant.

Cosecant Function

The cosecant function, denoted as \(\csc x\), is an important reciprocal trigonometric function. It is the inverse of the sine function, meaning it is defined as the reciprocal of sine. Mathematically, this is expressed as: \[ \csc x = \frac{1}{\sin x} \]Keep in mind that the cosecant function cannot be defined for values where \(\sin x = 0\) because division by zero is undefined. Hence, the cosecant function has vertical asymptotes at these points, such as \(x = 0, \pi, 2\pi\), etc. These asymptotes divide the cosecant graph into separate sections.As shown in the exercise, when \(x\) approaches \(\frac{\pi}{2}\) (a point where \(\sin x\) is not zero), \(\csc x\) is simply the reciprocal of 1, which also results in 1. The close association to the sine function makes it crucial to understand both functions together when evaluating limits.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions expand the basic trigonometric functions to accommodate more scenarios and to provide new insights by considering ratios in reverse. These include:

• Cosecant (\(\csc x = \frac{1}{\sin x}\)
  
• Secant (\(\sec x = \frac{1}{\cos x}\))
  
• Cotangent (\(\cot x = \frac{1}{\tan x}\))
  

The primary challenge with reciprocal functions is their undefined nature when the original trigonometric function is zero. For example, \(\csc x\) is undefined where \(\sin x = 0\). This leads to vertical asymptotes in the graph at those points, causing the function's values to diverge to infinity.Understanding reciprocal functions is essential when analyzing limits in calculus. In our exercise, knowing that \(\csc x\) is the reciprocal of \(\sin x\) provides direct insights into how the function behaves as \(x\) approaches critical values like \(\frac{\pi}{2}\). This dual perspective of both functions aids in a comprehensive understanding of trigonometric behavior at key points.