Problem 6 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) represents an angle measured in radians. $$ \csc x=2 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{6} + 2n\pi\) and \(x = \frac{5\pi}{6} + 2n\pi\), for any integer \(n\).

Step 1: Understanding the Cosecant Function

The cosecant function is the reciprocal of the sine function. Therefore, if \( \csc x = 2 \), then it means \( \frac{1}{\sin x} = 2 \). Rearranging this equation gives \( \sin x = \frac{1}{2} \).

Step 2: Finding the Principal Solution

The sine of an angle is \(\frac{1}{2}\) for angles \(x = \frac{\pi}{6}\) and \( x = \frac{5\pi}{6}\). These are the angles in the unit circle where the y-coordinate (sine value) is \(\frac{1}{2}\).

Step 3: Generating the General Solution

Trigonometric functions are periodic, which means they repeat their values over intervals. The sine function is periodic with period \(2\pi\). Therefore, the general solutions can be written as: \(x = \frac{\pi}{6} + 2n\pi\) and \(x = \frac{5\pi}{6} + 2n\pi\), where \(n\) is any integer.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc x \), is a fundamental trigonometric function. It is defined as the reciprocal of the sine function. In simpler terms, for a given angle \( x \), the cosecant is calculated as \( \csc x = \frac{1}{\sin x} \).
For the equation \( \csc x = 2 \), we can interpret this as needing the sine of the angle \( x \) to be \( \frac{1}{2} \).

• This relationship allows us to convert problems involving cosecant into more familiar sine problems.
• Understanding this reciprocal link forms the basis of solving many trigonometric equations.

Recognizing how the cosecant function ties back to sine is key to unlocking solutions in trigonometry.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. It is a crucial tool in trigonometry because it relates angles to the coordinates corresponding to those angles on the circle.

• On the unit circle, the x-coordinate of a point represents the cosine of the associated angle, while the y-coordinate represents the sine.
• For the angles where \( \sin x = \frac{1}{2} \), the corresponding angles are \( x = \frac{\pi}{6} \) and \( x = \frac{5\pi}{6} \).

By locating these angles on the unit circle, we identify where the y-coordinate reaches \( \frac{1}{2} \), thus offering visual clarity on the principal solutions.
This visual representation helps ensure retention and understanding of trigonometric solutions.

General Solution

In trigonometry, the general solution of an equation accounts for all possible values of the variable in question. Trigonometric functions like sine and cosecant are periodic, meaning they repeat their values in regular intervals.

• The sine function, for instance, has a period of \(2\pi\), meaning it repeats its values every \(2\pi\).
• So, if \( \sin x = \frac{1}{2} \) at \( x = \frac{\pi}{6} \) and \( x = \frac{5\pi}{6} \), it will repeat this at every additional multiple of \(2\pi\).

Consequently, the general solution to the equation \( \csc x = 2 \) becomes \( x = \frac{\pi}{6} + 2n\pi \) and \( x = \frac{5\pi}{6} + 2n\pi \), where \( n \) is any integer.
Understanding the concept of periodicity is essential to finding all solutions, not just the principal ones.