Problem 35 - Trigonometry

Question

The problems that follow review material we covered in Section 6.2. Find all solutions in the interval \(0^{\circ} \leq \theta<360^{\circ}\). If rounding is necessary, round to the nearest tenth of a degree. $$4 \sin \theta-\csc \theta=0$$

Step-by-Step Solution

Answer

The solutions are \(30^\circ, 150^\circ, 210^\circ,\) and \(330^\circ\).

Step 1: Rewrite the equation using trigonometric identities

The cosecant function can be rewritten in terms of the sine function, using the identity \( \csc \theta = \frac{1}{\sin \theta} \). Substitute this identity into the original equation to get: \( 4 \sin \theta - \frac{1}{\sin \theta} = 0 \).

Step 2: Clear the fraction

Multiply every term in the equation \( 4 \sin \theta - \frac{1}{\sin \theta} = 0 \) by \( \sin \theta \) to eliminate the fraction: \( 4 \sin^2 \theta - 1 = 0 \).

Step 3: Factor and solve for \( \sin \theta \)

Rearrange the equation to match a standard quadratic form: \( 4 \sin^2 \theta = 1 \). Divide both sides by 4 to simplify: \( \sin^2 \theta = \frac{1}{4} \). Take the square root of both sides: \( \sin \theta = \frac{1}{2} \) or \( \sin \theta = -\frac{1}{2} \).

Step 4: Find the angle solutions for \( \sin \theta = \frac{1}{2} \)

The angles where \( \sin \theta = \frac{1}{2} \) are \( \theta = 30^\circ \) and \( \theta = 150^\circ \) in the given interval \(0^\circ \leq \theta < 360^\circ\).

Step 5: Find the angle solutions for \( \sin \theta = -\frac{1}{2} \)

The angles where \( \sin \theta = -\frac{1}{2} \) are \( \theta = 210^\circ \) and \( \theta = 330^\circ \) in the given interval \(0^\circ \leq \theta < 360^\circ\).

Step 6: Compile all solutions

Gather all solutions found: \( \theta = 30^\circ, 150^\circ, 210^\circ, \) and \( 330^\circ \). These are all of the solutions for the equation in the specified interval.

Key Concepts

Cosecant Function

The cosecant function is one of the six primary trigonometric functions. It's closely related to the sine function, but with a twist. To understand the cosecant function, it's essential to grasp that it is the reciprocal of the sine function.

This means that if you know the value of sine for a particular angle, you can easily find the cosecant. The relationship is given as follows:

• The mathematical representation of the cosecant function is \( \csc \theta = \frac{1}{\sin \theta} \).

This identity is particularly helpful in solving trigonometric equations.

In the exercise we had, knowing this identity allowed us to convert the equation involving both sine and cosecant into one solely using the sine function. This is a common strategy when working with trigonometric equations, as it often simplifies the process and paves the way to finding solutions.

Quadratic Trigonometric Equations

Quadratic trigonometric equations are those that resemble the format of a regular quadratic equation, but instead involve trigonometric functions. These equations typically include squared terms and linear terms of the trigonometric functions.

In the exercise, once we used the trigonometric identity to express everything in terms of \( \sin \theta \), a quadratic form emerged:

• We obtained the equation \( 4 \sin^2 \theta - 1 = 0 \), which parallels familiar quadratic equations like \( ax^2 + bx + c = 0 \).

The strategy to solve this involves similar steps to regular quadratics such as factoring, applying the quadratic formula, or completing the square.

It's imperative to transform such equations to help recognize these patterns and apply appropriate methods. Solving the equation led us to discover values of \( \sin \theta \), showing how these techniques unfold solutions step by step.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the involved angles. These identities are crucial tools in both simplifying and solving trigonometric equations. They allow us to transform complex trigonometric expressions into simpler forms.

In the context of our exercise:

• We used the identity \( \csc \theta = \frac{1}{\sin \theta} \) to convert a complex initial equation into a simpler quadratic form.
• These identities help bridge the gap between different trigonometric functions, enabling easier manipulation of equations.

Having a strong grasp of these identities is invaluable for efficiently tackling trigonometric problems.

Mastering identities like the Pythagorean identities, reciprocal identities, and angle addition formulas, among others, equips you with a rich toolbox. This enhances problem-solving skills, making it easier to navigate through various trigonometric challenges.