Problem 32 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \csc (\theta / 3)=-1 $$

Step-by-Step Solution

Answer

The solution is \(\theta = 810^\circ + 1080^\circ n\), where \(n\) is any integer.

Step 1: Understanding the Equation

The given equation is \( \csc(\theta / 3) = -1 \). The cosecant function, \( \csc \), is the reciprocal of the sine function, so we have \( \csc(x) = \frac{1}{\sin(x)} \). This means we need to find when \( \sin(\theta / 3) = -1 \).

Step 2: Solving for the Sine Function

Since \( \sin(\theta / 3) = -1 \), we need to find the angles where the sine value equals \(-1\). The sine function is \(-1\) at \(270^\circ\) and repeats every \(360^\circ\). Thus, we have \( \theta / 3 = 270^\circ + 360^\circ n \), where \( n \) is any integer.

Step 3: Solving for Theta

Multiply both sides of the equation \( \theta / 3 = 270^\circ + 360^\circ n \) by 3 to isolate \( \theta \). This gives \( \theta = 810^\circ + 1080^\circ n \), where \( n \) is any integer. These are all the solutions for \( \theta \).

Step 4: Expressing the General Solution

The general solution of the equation \( \csc(\theta / 3) = -1 \) is \( \theta = 810^\circ + 1080^\circ n \), where \( n \) is any integer. This accounts for the periodic nature of the sine function.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc(x) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function. This means:

• \( \csc(x) = \frac{1}{\sin(x)} \)
• It's undefined when \( \sin(x) = 0 \) because division by zero is not possible.

The function is mainly used for computations where the sine value cannot be zero, otherwise, it can lead to undefined results.

In practice, the cosecant function helps convert problems involving the more commonly used sine function into a different format, but it isn't as frequently applied as sine. In this exercise, the transformation from \( \csc(\theta / 3) = -1 \) to \( \sin(\theta / 3) = -1 \) allowed us to directly find the solution using sine.

Sine Function

The sine function, represented as \( \sin(x) \), is another basic trigonometric function vital in the study of mathematics, physics, and engineering. Here's what to know:

• \( \sin(x) \) measures the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
• The function is periodic, repeating its values every \( 360^\circ \) (or \( 2\pi \) radians).

The sine function reaches its extrema at specific angles:

• Maximum at \( 90^\circ \), where \( \sin(x) = 1 \)
• Minimum at \( 270^\circ \), where \( \sin(x) = -1 \)

In our problem, solving \( \sin(\theta / 3) = -1 \) gives the angle \( \theta / 3 = 270^\circ + 360^\circ n \). This equation leveraged the sine function's periodic property to provide a comprehensive solution.

Angles in Degrees

Angles are measured in degrees within the context of trigonometry and geometry. Understanding degrees is essential when working with trigonometric equations.

• A full circle is divided into \( 360 \) degrees.
• Each right angle is \( 90^\circ \).

Degrees allow us to express angles in familiar terms, like quarters or thirds of a circle. For trigonometric functions, key angle measures in degrees include \( 0^\circ, 90^\circ, 180^\circ, 270^\circ, \) and full circle at \( 360^\circ \).

The periodicity of trigonometric functions like sine and cosecant, closely ties to angles in degrees. In this exercise, knowing the angle \( \theta / 3 = 270^\circ + 360^\circ n \) helps identify when the sine (and thus cosecant) values repeat themselves. By solving in degrees, we ensured the solutions were straightforward interpretations of the problem. This understanding allowed us to find \( \theta = 810^\circ + 1080^\circ n \) by simply multiplying by 3.