Problem 18 - Algebra and Trigonometry

Question

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ 3 \csc ^{2} \theta=4 $$

Step-by-Step Solution

Answer

The solutions are \(\frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\).

Step 1: Understanding the Given Equation

We start with the given equation: \(3 \csc^{2} \theta = 4\). The cosecant function is the reciprocal of the sine function, so \(\csc \theta = \frac{1}{\sin \theta}\). Therefore, \(\csc^{2} \theta = \frac{1}{\sin^{2} \theta}\). Rewrite the equation as \(3 \times \frac{1}{\sin^2 \theta} = 4\).

Step 2: Isolate \(\sin^2 \theta\)

Multiply both sides of the equation \(3 \times \frac{1}{\sin^2 \theta} = 4\) by \(\sin^2 \theta\) and divide by 4 to isolate \(\sin^2 \theta\): \(\sin^2 \theta = \frac{3}{4}\).

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

Take the square root of both sides to solve for \(\sin \theta\): \(\sin \theta = \pm \sqrt{\frac{3}{4}}\). Simplifying further gives \(\sin \theta = \pm \frac{\sqrt{3}}{2}\).

Step 4: Find All Angles

The general solutions for \(\sin \theta = \frac{\sqrt{3}}{2}\) are \(\theta = \frac{\pi}{3} + 2k\pi\) and \(\theta = \frac{2\pi}{3} + 2k\pi\). For \(\sin \theta = -\frac{\sqrt{3}}{2}\), the solutions are \(\theta = \frac{4\pi}{3} + 2k\pi\) and \(\theta = \frac{5\pi}{3} + 2k\pi\), where \(k\) is any integer.

Step 5: Find Solutions in \\[0, 2\pi)\\

Restrict the solutions from Step 4 to the interval \([0, 2\pi)\). \(\theta = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3},\) and \(\frac{5\pi}{3}\) are all within \([0, 2\pi)\).

Key Concepts

Cosecant Function

The cosecant function, often denoted as \( \csc \theta \), is the reciprocal of the sine function. This means that for any angle \( \theta \), the cosecant is defined as \( \csc \theta = \frac{1}{\sin \theta} \). This function is particularly useful in situations where sine is zero because it would cause the reciprocal to be undefined.

• The cosecant function is not defined for angles where \( \sin \theta = 0 \), such as \( \theta = 0, \pi, 2\pi, \ldots \)
• It shares the same periodic nature as the sine function, repeating its values every \(2\pi\) radians.

Understanding the cosecant function is crucial when solving trigonometric equations like the one presented in the exercise. The equation \( 3 \csc^{2} \theta = 4 \) involves manipulating the cosecant function, specifically converting it into terms of sine for easier handling.

Sine Function

The sine function, represented as \( \sin \theta \), is a fundamental trigonometric function that relates an angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. It is one of the basic building blocks of trigonometry, and it oscillates between -1 and 1.

• The sine function has a period of \(2\pi\), meaning it repeats its pattern every \(2\pi\) radians.
• The sine of an angle \( \theta \) is positive in the first and second quadrants and negative in the third and fourth quadrants.

In the context of the example given, understanding \( \sin \theta \) was essential to solve for \( \csc^{2} \theta \) as it required expressing the equation in terms of \( \sin \theta \) before finding specific angle solutions. For the equation \( 3 \csc^{2} \theta = 4 \), it transformed to \( \sin^2 \theta = \frac{3}{4} \), inviting the use of sine's properties to find all potential values of \( \theta \).

General Solutions

General solutions in trigonometry refer to expressing all possible angles that satisfy a specific trigonometric equation. Since trigonometric functions are periodic, one initial solution can lead to infinitely many others by adding the period of the function. For example, the sine function has a period of \(2\pi\), so solutions can be expressed in terms of \(\theta + 2k\pi\), where \(k\) is an integer.

• For \( \sin \theta = \frac{\sqrt{3}}{2} \), general solutions are \( \theta = \frac{\pi}{3} + 2k\pi \) and \( \theta = \frac{2\pi}{3} + 2k\pi \).
• For \( \sin \theta = -\frac{\sqrt{3}}{2} \), general solutions are \( \theta = \frac{4\pi}{3} + 2k\pi \) and \( \theta = \frac{5\pi}{3} + 2k\pi \).

The ability to derive general solutions is powerful as it systematically accounts for all angle rotations that result in the same sine value. In solving trigonometric equations, it ensures that no solutions are missed in the process.

Radians

Radians are a unit of measure for angles used predominantly in mathematics. Unlike degrees, which divide a circle into 360 parts, radians express angles as the ratio of the arc length to the radius. One full rotation around a circle is \(2\pi\) radians.

• Most trigonometric equations are solved in radians due to their natural mathematical properties.
• Many basic angles have simple radian equivalents: \( \pi \) radians for 180 degrees and \( \frac{\pi}{2} \) radians for 90 degrees.

When finding solutions to trigonometric equations, such as restricting \( \theta \) to the interval \([0, 2\pi)\), radians offer a seamless transition between cycling periods of trigonometric functions and their solutions. It helps maintain consistency and simplify calculations, especially when finding solutions for broader or restricted intervals.