Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ \csc \gamma=\sqrt{2} $$

Step-by-Step Solution

Answer

Solutions: \( \gamma = \frac{\pi}{4} + 2k\pi \) and \( \gamma = \frac{3\pi}{4} + 2k\pi \), \( k \in \mathbb{Z} \).

Step 1: Understand the function

The function \( \csc \gamma \) represents the cosecant of \( \gamma \), which is defined as \( \frac{1}{\sin \gamma} \). Our task is to find \( \gamma \) such that \( \csc \gamma = \sqrt{2} \).

Step 2: Rewrite the equation in terms of sine

Rewriting \( \csc \gamma = \sqrt{2} \) in terms of sine, we have:\[ \frac{1}{\sin \gamma} = \sqrt{2} \]Taking the reciprocal of both sides gives:\[ \sin \gamma = \frac{1}{\sqrt{2}} \]

Step 3: Simplify \( \sin \gamma \) values

We further simplify \( \sin \gamma = \frac{1}{\sqrt{2}} \) to \( \sin \gamma = \frac{\sqrt{2}}{2} \), which is a standard sine value.

Step 4: Find general solutions for \( \gamma \)

The angle \( \gamma \) can be found from the unit circle or using standard angle values. The solutions to \( \sin \gamma = \frac{\sqrt{2}}{2} \) are:\[ \gamma = \frac{\pi}{4} + 2k\pi \quad \text{and} \quad \gamma = \frac{3\pi}{4} + 2k\pi \]where \( k \) is any integer.

Step 5: State the final solution set

The complete set of solutions for the equation \( \csc \gamma = \sqrt{2} \) is:\[ \gamma = \frac{\pi}{4} + 2k\pi \quad \text{and} \quad \gamma = \frac{3\pi}{4} + 2k\pi \] where \( k \) is any integer, representing all angles whose sine value is \( \frac{\sqrt{2}}{2} \).

Key Concepts

Trigonometric Equations

Trigonometric equations are mathematical statements that involve trigonometric functions like sine, cosine, and their inverses. These equations are used to find angles whose trigonometric ratios satisfy given conditions. For instance, in the original exercise, the equation \( \csc \gamma = \sqrt{2} \) is a trigonometric equation. The cosecant function (\( \csc \gamma \)) is the reciprocal of the sine function, so solving the equation involves converting it into terms of a sine function and finding the angles that satisfy this condition.
Solving trigonometric equations typically involves several steps:

• Identify the trigonometric function involved. Here, it's the cosecant function.
• Rewrite the equation using basic trigonometric identities or algebraic manipulations. For example, turning \( \csc \gamma = \sqrt{2} \) to \( \sin \gamma = \frac{\sqrt{2}}{2} \).
• Determine solutions using known angles and patterns, often utilizing tools like the unit circle or known values of trigonometric functions.

The goal is always to find all possible angles that make the equation true within its domain, and often, these solutions are represented in terms of general expressions like \( \gamma = \frac{\pi}{4} + 2k\pi \), where \( k \) is an integer.

Unit Circle

The unit circle is an essential tool in trigonometry. It helps solve trigonometric equations by providing a visual representation of angles and their corresponding sine and cosine values. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system.
The coordinates on the unit circle correspond to the cosine and sine of the angle formed with the positive x-axis. For example:

• At angle \( \frac{\pi}{4} \), the coordinates are \( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \).
• At angle \( \frac{3\pi}{4} \), the coordinates are \( \left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \).

Since cosecant is the reciprocal of sine, when you need to solve an equation like \( \csc \gamma = \sqrt{2} \), translating it to \( \sin \gamma = \frac{\sqrt{2}}{2} \) allows us to identify the angles \( \frac{\pi}{4} \) and \( \frac{3\pi}{4} \) on the unit circle where these values occur. These angles provide reference points from which additional solutions can be deduced using periodicity.

Sine Function

The sine function is a fundamental trigonometric function. It represents the y-coordinate of the point where the terminal side of an angle intersects the unit circle. The sine function is periodic, with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians.
Key properties of the sine function include:

• Range: Sine values lie between -1 and 1.
• Symmetry: The sine function is odd, meaning \( \sin(-x) = -\sin(x) \).
• Key angles: Commonly used values are \( \sin(0) = 0 \), \( \sin\left(\frac{\pi}{2}\right) = 1 \), and \( \sin(\pi) = 0 \).

When solving \( \sin \gamma = \frac{\sqrt{2}}{2} \), we identify specific angles \( \gamma = \frac{\pi}{4} \) and \( \gamma = \frac{3\pi}{4} \) because these are familiar angles where the sine value matches \( \frac{\sqrt{2}}{2} \). By exploiting the periodic nature of the sine function, we can express solutions as infinite series \( \gamma = \frac{\pi}{4} + 2k\pi \) and \( \gamma = \frac{3\pi}{4} + 2k\pi \), ensuring all potential solutions are captured.