Problem 11 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) represents a real number. $$ -\csc x=1 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{3\pi}{2} + 2k\pi\), where \(k\) is any integer.

Step 1: Rewrite the Equation

The given equation is \(-\csc x = 1\). To solve this, first multiply both sides by -1 to obtain \(\csc x = -1\).

Step 2: Recall Cosecant's Definition

The cosecant function is defined as \(\csc x = \frac{1}{\sin x}\). So, \(\csc x = -1\) implies \(\frac{1}{\sin x} = -1\).

Step 3: Solve for \(\sin x\)

To solve \(\frac{1}{\sin x} = -1\), take the reciprocal of both sides. This gives \(\sin x = -1\).

Step 4: Identify Angles that Satisfy \(\sin x = -1\)

\(\sin x = -1\) occurs when \(x = \frac{3\pi}{2} + 2k\pi\), where \(k\) is any integer, because the sine function equals -1 at these angles on the unit circle.

Step 5: Express the General Solution

The general solution for \(x\) is thus \(x = \frac{3\pi}{2} + 2k\pi\), where \(k\) is any integer. This accounts for all possible solutions corresponding to the given equation.

Key Concepts

Cosecant Function

The cosecant function is a trigonometric function that is the reciprocal of the sine function. This means that cosecant, denoted as \(\csc x\), is defined by the equation:

• \(\csc x = \frac{1}{\sin x} \)

This definition implies that wherever the sine function is zero, the cosecant function becomes undefined, as division by zero is not possible. The cosecant function is particularly useful in solving trigonometric equations, especially when expressing it in terms of other trigonometric functions, like sine.

Sine Function

The sine function is a fundamental trigonometric function represented as \(\sin x\). It arises from the unit circle, which we'll discuss later. The sine function maps angles to their respective values on a vertical axis of the unit circle. The values of the sine function range from -1 to 1.
This function reveals significant information in solving trigonometric equations:

• It helps us find angles corresponding to certain sine values.
• It provides the basis for understanding how other trigonometric functions behave.

When \( \sin x = -1 \), it means the angle is in a specific position on the unit circle, leading us to possible solutions for trigonometric equations.

Unit Circle

The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. This circle helps visualize the sine and cosine functions. It shows all possible values of these trigonometric functions as points on the circle, correlating angle measures with their respective sine and cosine values.
When working with trigonometric equations like \(\sin x = -1\), the unit circle is invaluable in determining the angle that corresponds to this value:

• \( \sin x = -1 \) appears at the angle \(\frac{3\pi}{2} \), pointing straight down on the circle.

Therefore, the unit circle is crucial in identifying where specific trigonometric values are met.

General Solution

The general solution in trigonometry refers to all possible solutions that satisfy a given equation across all cycles of the trigonometric functions. Since trigonometric functions are periodic, they repeat values over intervals known as periods. For sine and cosine, this period is \(2\pi\).
In our equation where \(\sin x = -1\), the angle \(\frac{3\pi}{2} \) is a specific solution. However, it can repeat every \( 2\pi \) radians, providing an infinite set of solutions:

• The general solution formula is \(x = \frac{3\pi}{2} + 2k\pi\), where \( k \) is any integer.

This formula ensures every occurrence of the sine of \(x\) equaling to -1 is captured, representing the complete set of solutions for the equation.