Problem 39 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\) $$ \csc \theta=-1 $$

Step-by-Step Solution

Answer

The solution to the equation \( \csc \theta = -1 \) for \(0 \leq \theta <2 \pi\) is \( \theta = 3\pi/2 \).

Step 1: Understand the Cosecant Function

The cosecant function is defined as \( \csc \theta = 1/ \sin \theta \). So, when \( \csc \theta = -1 \), it means \( \sin \theta = -1 \).

Step 2: Determine the Angle

From the unit circle, it is known that \( \sin \theta = -1 \) when \( \theta = 3\pi/2 \).

Step 3: Check the Range

Since the angle \( \theta = 3\pi/2 \) falls within the range \( 0 \leq \theta < 2\pi \), this is the solution to the given equation.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is a trigonometric function which is the reciprocal of the sine function. This means that \( \csc \theta = \frac{1}{\sin \theta} \).
When you encounter an equation involving the cosecant function, such as \( \csc \theta = -1 \), it is important to remember that you can rewrite it in terms of the sine function: \( \sin \theta = -1 \).
Transforming the equation like this often makes it easier to solve, especially when you use familiar values from the unit circle.

• Keep in mind that the cosecant function is undefined when \( \sin \theta = 0 \) because division by zero is not possible.
• The graph of the cosecant function is closely related to that of the sine function, featuring the same periodicity and zero-crossings where it is undefined.

Understanding the behavior of the cosecant function is essential for solving trigonometric equations, especially those in the form \( \csc \theta = k \).

Unit Circle

The unit circle is a powerful tool for solving trigonometric equations, especially when you are trying to determine angles that satisfy a particular trigonometric function. In the context of the exercise, the unit circle helps us find where \( \sin \theta = -1 \).
The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. On this circle, each angle \( \theta \) corresponds to a point, where the x-coordinate is \( \cos \theta \) and the y-coordinate is \( \sin \theta \).

• The full circle extends from \( 0 \) radians (or degrees) to \( 2\pi \) radians (360 degrees).
• The angle \( \theta = \frac{3\pi}{2} \) is located at the bottom of the unit circle, where the coordinates are (0, -1).

At this angle, \( \sin \theta \) is indeed \(-1\), solving the original problem. Using the unit circle allows us to visually and logically find solutions to trigonometric equations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities are essential tools for manipulating and solving trigonometric equations. Some key identities include:

• Reciprocal identities, such as \( \csc \theta = \frac{1}{\sin \theta} \)
• Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Angle sum and difference identities, like \( \sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \)

In the context of the problem, recognizing the reciprocal identity for \( \csc \theta \) is crucial as it allows you to transform the equation into something more straightforward involving \( \sin \theta \).
By using these identities, one can consistently solve complex trigonometric problems by transforming them into simpler forms or looking up related values. Building a strong grasp on these identities will make solving trigonometric equations much easier.