Problem 44 - Precalculus Mathematics for Calculus

Question

Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi)\) $$\tan \frac{\theta}{2}-\sin \theta=0$$

Step-by-Step Solution

Answer

Solutions are \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\).

Step 1: Recognize the equation format

The equation given is \(\tan \frac{\theta}{2} - \sin \theta = 0\). The terms involve trigonometric functions of different types, indicating that a formula might be needed to relate these functions.

Step 2: Apply the double angle formula

Use the identity for \(\tan \frac{\theta}{2}\), which is \(\tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta}\). Substitute this into the original equation to get \(\frac{\sin \theta}{1 + \cos \theta} - \sin \theta = 0\).

Step 3: Simplify the equation

Combine the terms over a common denominator: \(\frac{\sin \theta - \sin \theta (1 + \cos \theta)}{1 + \cos \theta} = 0\). This simplifies to \(\frac{\sin \theta - \sin \theta - \sin \theta \cos \theta}{1 + \cos \theta} = \frac{-\sin \theta \cos \theta}{1 + \cos \theta} = 0\).

Step 4: Solve the simplified equation

Since the numerator must be zero, set \(-\sin \theta \cos \theta = 0\). This implies \(\sin \theta \cos \theta = 0\).

Step 5: Analyze the solutions

From \(\sin \theta \cos \theta = 0\), either \(\sin \theta = 0\) or \(\cos \theta = 0\). Solving these equations in the interval \([0, 2\pi)\):- If \(\sin \theta = 0\), \(\theta = 0, \pi\).- If \(\cos \theta = 0\), \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\).

Step 6: List the solutions

The solutions for \(\theta\) within the interval \([0, 2\pi)\) are \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\).

Key Concepts

Trigonometric Equations

Trigonometric equations involve expressions containing trigonometric functions like sine, cosine, and tangent. These equations often require special techniques for solving, due to the periodic nature of the functions involved. Understanding trigonometric identities and formulas is essential to manipulate and solve these equations effectively. For instance, consider our exercise, where the original equation is \(\tan \frac{\theta}{2} - \sin \theta = 0\). This type of equation signals the need for applying a trigonometric identity to transform it into a solvable form.When solving trigonometric equations, some key steps include:

• Recognizing trigonometric identities that can simplify the equation.
• Transforming the equation using these identities or formulas.
• Simplifying the resulting expressions to isolate the variable on one side.

Performing these steps correctly allows finding solutions in the given interval \([0, 2\pi)\). The periodicity of trigonometric functions may result in multiple solutions within a specified interval.

Half-Angle Formula

The Half-Angle Formula is a useful trigonometric identity for simplifying expressions involving half angles. Particularly, the identity for the tangent of a half-angle, \(\tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta}\), comes in handy when dealing with trigonometric equations where direct computation is challenging. In our exercise, the half-angle formula helped convert \(\tan \frac{\theta}{2}\) into a form compatible with \(\sin \theta\). By substituting \(\frac{\sin \theta}{1 + \cos \theta}\) for \(\tan \frac{\theta}{2}\) in the equation \(\tan \frac{\theta}{2} - \sin \theta = 0\), we establish a single trigonometric function equation, facilitating simplification and solving. Using the half-angle formulas can:

• Simplify complex expressions involving double or half angles.
• Provide a path to rewrite equations in simpler, more solvable terms.
• Enable the solving of equations in specified intervals, by converting them to relate directly known trigonometric functions.

Thus, understanding and applying the half-angle formulas broadens one's toolkit for solving trigonometric equations.

Interval Analysis

Interval analysis in trigonometry refers to evaluating solutions of trigonometric equations within a predefined range. In our exercise, we were given the interval \([0, 2\pi)\). The solutions must therefore be determined within this specific interval, ensuring all possible angles are considered. In this context, it's important to:

• Identify all solutions of the simplified equation \(\sin \theta \cos \theta = 0\).
• Ensure that all solutions fall within the given interval \([0, 2\pi)\).
• Acknowledge the periodicity of trigonometric functions which can produce multiple solutions within the interval.

For instance, \(\sin \theta = 0\) leads to \(\theta = 0, \pi\), whereas \(\cos \theta = 0\) results in \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\) within this range. By applying interval analysis, we ensure the solution set is relevant and complete for the given constraints. Thus, enabling us to provide the final set of solutions: \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\).