Problem 53 - Thomas Calculus

Question

For Exercises \(51-54,\) solve for the angle \(\theta,\) where \(0 \leq \theta \leq 2 \pi\). $$\sin 2 \theta-\cos \theta=0$$

Step-by-Step Solution

Answer

The solutions are \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{6}, \frac{5\pi}{6}\).

Step 1: Use Double Angle Identity

The given equation is \(\sin 2 \theta - \cos \theta = 0\). We can use the double angle identity for sine, which is \(\sin 2\theta = 2 \sin \theta \cos \theta\). Substituting this into the equation gives us: \(2 \sin \theta \cos \theta - \cos \theta = 0\).

Step 2: Factor the Equation

We can factor the equation \(2 \sin \theta \cos \theta - \cos \theta = 0\) by taking \(\cos \theta\) as a common factor, resulting in: \(\cos \theta(2 \sin \theta - 1) = 0\).

Step 3: Use Zero Product Property

Using the zero product property, solve each factor: \(\cos \theta = 0\) and \(2 \sin \theta - 1 = 0\).

Step 4: Solve \(\cos \theta = 0\)

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

Step 5: Solve \(2 \sin \theta - 1 = 0\)

Solve \(2 \sin \theta - 1 = 0\) by first adding 1 to both sides and then dividing by 2, we get: \(\sin \theta = \frac{1}{2}\).

Step 6: Find Solutions for \(\sin \theta = \frac{1}{2}\)

Determine the angles \(\theta\) where \(\sin \theta = \frac{1}{2}\) within \([0, 2\pi]\). These angles are \(\theta = \frac{\pi}{6}, \frac{5\pi}{6}\).

Step 7: Compile All Solutions

Combine all solutions from Steps 4 and 6. Thus, the complete set of solutions for \(\theta\) is: \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{6}, \frac{5\pi}{6}\).

Key Concepts

Double Angle Identities

In trigonometry, double angle identities play a crucial role in simplifying and solving equations. One of the most common double angle identities is for the sine function: \( \sin 2\theta = 2 \sin \theta \cos \theta \). This identity is a powerful tool because it allows us to express trigonometric functions involving double angles in terms of single angles.

To understand why this identity is useful, consider our exercise: \( \sin 2 \theta - \cos \theta = 0 \). By substituting the double angle identity for sine, \( \sin 2 \theta \) is replaced with \( 2 \sin \theta \cos \theta \), transforming the equation into something more manageable: \( 2 \sin \theta \cos \theta - \cos \theta = 0 \). This transformation simplifies the process of solving the equation since it moves from a double angle back to expressions that can be factored easily.

Zero Product Property

The zero product property is a fundamental principle that helps in solving equations that have been factored. It states that if the product of two quantities is zero, then at least one of the quantities must be zero.

In the context of our equation \( \cos \theta (2 \sin \theta - 1) = 0 \), the zero product property tells us that either \( \cos \theta = 0 \) or \( 2 \sin \theta - 1 = 0 \) (or both) must be zero for the product to be zero.

• To solve \( \cos \theta = 0 \), we look for angles \( \theta \) within the specified interval \([0, 2\pi]\) where the cosine value is zero.
• For \( 2 \sin \theta - 1 = 0 \), we rearrange it to \( \sin \theta = \frac{1}{2} \), then find the corresponding angles \( \theta \).

Understanding and applying the zero product property effectively allows us to break down complex equations into simpler, more workable components.

Angle Solutions

Finding angle solutions involves determining specific angles within a given range that satisfy the trigonometric equations. This requires knowledge of the unit circle and the properties of sine and cosine functions.

For the equation \( \cos \theta = 0 \), the angle solutions within \([0, 2\pi]\) are \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \). These angles correspond to the points on the unit circle where the x-coordinate (which represents cosine) is zero.

• \( \theta = \frac{\pi}{2} \) is located at the top of the unit circle, and
• \( \theta = \frac{3\pi}{2} \) is at the bottom of the unit circle.

Solving \( \sin \theta = \frac{1}{2} \) requires finding angles where the y-coordinate (representing sine) equals \( \frac{1}{2} \). Within the interval \([0, 2\pi]\), these angles are \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{5\pi}{6} \).

By combining all these solutions, we can fully solve the original exercise and provide the complete set of angles that satisfy the equation: \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{6}, \frac{5\pi}{6} \).