Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\csc 3 x $$

Step-by-Step Solution

Answer

The period is \( \frac{2\pi}{3} \) and the vertical asymptotes are at \( x = \frac{n\pi}{3} \), \( n \in \mathbb{Z} \).

Step 1: Understand the Function

The function given is the cosecant function, which is the reciprocal of the sine function. So, \( y = \csc 3x = \frac{1}{\sin 3x} \). This means the function is undefined when \( \sin 3x = 0 \). This will help us find the asymptotes.

Step 2: Find the Basic Period

The basic period of \( \csc x = \frac{1}{\sin x} \) is the same as that of the sine function, which is \( 2\pi \). However, due to the coefficient of 3 inside the function, the period is \( \frac{2\pi}{3} \). This is because the function \( \sin(kx) \) has a period of \( \frac{2\pi}{k} \).

Step 3: Determine the Vertical Asymptotes

Vertical asymptotes occur where the sine function is equal to zero, i.e., at \( 3x = n\pi \), where \( n \) is an integer. Solving for \( x \), we get \( x = \frac{n\pi}{3} \). These are the vertical asymptotes: \( x = \frac{n\pi}{3} \), where \( n \) is any integer.

Step 4: Sketching one period of the graph

On a graph, between each pair of asymptotes, the cosecant function will have a section that resembles half of an 'U' or 'n' shape, depending on whether the sine is positive or negative. For one period \( [0, \frac{2\pi}{3}] \), draw vertical lines (asymptotes) at \( x = 0 \) and \( x = \frac{2\pi}{3} \). The cosecant function will have a 'n'-shaped section between these lines, peaking or valleying midway.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are fundamental in mathematics, primarily dealing with angles and the relationships between side lengths in right-angled triangles. These functions are sine, cosine, tangent, cosecant, secant, and cotangent. Each has a distinct relationship and usage:

• **Sine (sin)** measures the ratio of the opposite side to the hypotenuse.
• **Cosine (cos)** is the ratio of the adjacent side to the hypotenuse.
• **Tangent (tan)**, the ratio of the opposite side to the adjacent side.
• **Cosecant (csc)** is the reciprocal of sine, or \( \csc(x) = \frac{1}{\sin(x)} \).
• **Secant (sec)** is the reciprocal of cosine.
• **Cotangent (cot)** is the reciprocal of tangent.

The cosecant function, as highlighted in the problem, can express properties about wave-like phenomena in physics and engineering, due to its periodic nature.

Graphing Asymptotes in Trigonometric Functions

Asymptotes are crucial in graphing trigonometric functions like the cosecant function. These are lines that the graph approaches but never touches. For functions such as \( y = \csc(3x) \), asymptotes occur where the sine function (the denominator) is zero, making the function undefined.Determining the precise location of asymptotes involves solving equations where the denominator equals zero:

• For \( y = \csc(3x) \), this means finding where \( \sin(3x) = 0 \).
• Solving this gives us \( 3x = n\pi \) with \( n \) being an integer.
• Upon dividing by 3, we find our asymptotes at \( x = \frac{n\pi}{3} \).

When graphing, these are plotted as vertical lines, showing where the graph is not defined. This helps create the characteristic 'U' or 'n' shape sections of the cosecant curve between these lines.

Period of Trigonometric Functions

The period of a trigonometric function is the interval after which the function repeats its values. This concept is particularly relevant in functions involving sine and cosine, which naturally have a period of \( 2\pi \).When dealing with a modified function like \( y = \csc(3x) \):

• The basic period of \( \csc(x) \), similar to the sine function, is \( 2\pi \).
• But the coefficient 3 inside the function \( \sin(3x) \) alters the period.
• The formula for finding the new period is \( \frac{2\pi}{k} \), where \( k \) is the coefficient of \( x \).
• Thus, the period of \( \csc(3x) \) is \( \frac{2\pi}{3} \).

Understanding the period is essential when sketching the graph, as it determines the length of one complete cycle from start to end. Identifying this interval precisely ensures the accuracy of graphical representation.