Problem 71 - Algebra and Trigonometry

Question

Find all solutions of the equation that lie in the interval \([0, \pi] .\) State each answer correct to two decimal places. $$ \csc x=3 $$

Step-by-Step Solution

Answer

Solutions are approximately 0.34 and 2.80 radians.

Step 1: Understand the Cosecant Function

The cosecant function, \( \csc x \), is the reciprocal of the sine function, meaning \( \csc x = \frac{1}{\sin x} \). Thus, our equation \( \csc x = 3 \) can be rewritten in terms of sine as \( \sin x = \frac{1}{3} \).

Step 2: Solve for Sine

We must find \( x \) such that \( \sin x = \frac{1}{3} \). The sine function is positive in the interval \([0, \pi]\), specifically in the first quadrant and approaches zero at \( x = \pi \).

Step 3: Find the First Quadrant Solution

The sine function is increasing in the first quadrant \([0, \frac{\pi}{2}]\). Use the inverse sine function to find the angle: \( x = \arcsin\left(\frac{1}{3}\right) \). Calculating this gives \( x \approx 0.34 \) radians.

Step 4: Consider the Second Quadrant Solution

In the interval \([0, \pi]\), the sine function is symmetric around \( \frac{\pi}{2} \). The second quadrant solution can be found by reflecting the first quadrant angle over \( \frac{\pi}{2} \): \( x = \pi - 0.34 \approx 2.80 \) radians.

Step 5: Verify Solutions

Double-check that both solutions \( x \approx 0.34 \) and \( x \approx 2.80 \) lie within the interval \([0, \pi]\) and satisfy the original equation \( \csc x = 3 \).

Key Concepts

Cosecant Function

The cosecant function is part of a family of trigonometric functions which includes sine, cosine, tangent, secant, and cotangent. Specifically, the cosecant function is the reciprocal of the sine function. This means that for any angle \( x \), \( \csc x = \frac{1}{\sin x} \). Understanding this relationship is crucial for solving equations involving cosecant.

In our exercise, we start with the equation \( \csc x = 3 \). By expressing the cosecant in terms of sine, we convert the equation into \( \sin x = \frac{1}{3} \). This conversion simplifies the process of finding solutions using standard trigonometric techniques.

Key points to remember about the cosecant function are:

• Cosecant is undefined whenever the sine is zero, so it's important to understand the behavior of sine across its domain.
• The range of cosecant is \( (-\infty, -1] \cup [1, \infty) \), so values must be outside of the \((-1, 1)\) interval.
• Like sine, the periodicity of the cosecant is \( 2\pi \).

Inverse Sine Function

The inverse sine function, denoted as \( \arcsin(x) \), is used to find an angle \( x \) when the sine of the angle is known. In mathematical terms, if \( \sin y = x \), then \( y = \arcsin(x) \). This function is particularly useful when solving trigonometric equations where an angle measurement is required.

In our task, after transforming the cosecant function into a sine equation, \( \sin x = \frac{1}{3} \), we use the inverse sine function to find the angle. This involves calculating \( x = \arcsin(\frac{1}{3}) \), giving us a value of approximately 0.34 radians.

The inverse sine function has specific properties:

• It is defined only for \( x \) values within the interval \([-1, 1]\), which matches the range of the sine function.
• The output range of \( \arcsin(x) \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), which means it provides angles within this range.
• It’s useful for finding principal values of sine that help determine full sets of solutions for any given problem.

Understanding these characteristics allows us to find accurate solutions within specified problem intervals.

Trigonometric Identities

Trigonometric identities are equations that are true for all angles and are instrumental in simplifying and solving trigonometric equations. They serve as the foundation for manipulating trigonometric expressions to find solutions.

In this problem, understanding how the sine function behaves in different quadrants helps us find all relevant solutions. The sine function is positive in both the first and second quadrants within the interval \([0, \pi]\). Utilizing the symmetry property of sine, any angle \( x \) in the first quadrant will have a corresponding angle \( \pi - x \) in the second quadrant.

When solving \( \csc x = 3 \), the solution \( x = \arcsin(\frac{1}{3}) \) gives us our first angle, approximately 0.34 radians in the first quadrant. By reflecting this angle across \( \frac{\pi}{2} \) to find the second quadrant equivalent, we use the identity \( x = \pi - \arcsin(\frac{1}{3}) \), resulting in approximately 2.80 radians.

It's crucial to remember these key trigonometric identities:

• Reciprocal identities (e.g., \( \csc x = \frac{1}{\sin x} \)).
• Pythagorean identities.
• Symmetry identities for reflecting angles in different quadrants.

Mastery of trigonometric identities enhances the ability to solve equations and understand the underlying principles of trigonometry.