Problem 45 - Algebra 2

Question

Use identities to write each equation in terms of the single angle \(\theta .\) Then solve the equation for \(0 \leq \theta<2 \pi .\) $$ \cos 2 \theta=-2 \cos ^{2} \theta $$

Step-by-Step Solution

Answer

The solutions are \( \theta = \pi/3, 2 \pi/3, 4 \pi/3, 5 \pi/3 \).

Step 1: Use the Double-Angle Identity for Cosine

The double-angle identity for cosine is \( \cos 2 \theta = 1 - 2 \sin^2 \theta \) or \( \cos 2 \theta = 2 \cos^2 \theta - 1 \). For this exercise, the second form of the identity makes the equation easier to solve. So, rewrite the equation using the identity \( \cos 2 \theta = 2 \cos^2 \theta - 1 \), you have: \( 2 \cos^2 \theta - 1 = -2 \cos ^2\theta \).

Step 2: Simplify the equation

Combine like terms to simplify the equation. Add \( 2 \cos ^2\theta \) to both sides to get \( 4 \cos^2 \theta = 1 \). Then, divide by 4 to isolate \( \cos^2 \theta \) which gives: \( \cos^2 \theta = \frac{1}{4} \).

Step 3: Use the identity for Cosine Squared

The identity for \( \cos^2 \theta \) is \( \cos \theta = \pm \sqrt{\cos^2 \theta} \). Therefore, \( \cos \theta = \pm \frac{1}{2} \).

Step 4: Solve for theta

Using the unit circle or cosine table, solve for \( \theta \) where \( 0 \leq \theta < 2 \pi \). The solutions are \( \theta = \pi/3, 5 \pi/3 \) for \( \cos \theta = 1/2 \) and \( \theta = 2 \pi/3, 4 \pi/3 \) for \( \cos \theta = -1/2 \).

Key Concepts

Double-Angle Identity

The double-angle identity is a trigonometric formula that helps simplify expressions where angles are doubled. In the context of the cosine function, it allows us to express \( \cos 2\theta \) in two main forms:

• \( \cos 2\theta = 1 - 2 \sin^2 \theta \)
• \( \cos 2\theta = 2 \cos^2 \theta - 1 \)

For the problem at hand, we use the second form: \( \cos 2 \theta = 2 \cos^2 \theta - 1 \). This choice is convenient because it directly relates \( \cos 2\theta \) to the square of \( \cos \theta \), making it easier to rewrite and solve the original equation for \( \theta \). Understanding these identities simplifies our work by providing alternative expressions to tackle complex trigonometric equations effectively.

Cosine Function

The cosine function, denoted as \( \cos \theta \), is one of the fundamental trigonometric functions. It relates the angle \( \theta \) to the x-coordinate of a point on the unit circle. Specifically, the cosine of any angle \( \theta \) is the horizontal distance from the origin to the point on the unit circle at angle \( \theta \).
This function is periodic, with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians. This periodic nature is vital when solving trigonometric equations, as it helps identify all possible solutions within a specified interval. In this exercise, recognizing that \( \cos(\theta) = \pm 1/2 \) allows us to explore multiple angles within a \( 0 \leq \theta < 2\pi \) range.

Unit Circle

The unit circle is a circle with a radius of one. It's a powerful tool for visualizing and understanding trigonometric functions like sine and cosine. Each point on the unit circle corresponds to an angle \( \theta \), with the coordinates \((\cos \theta, \sin \theta)\) giving the cosine and sine values for that angle.
To solve the equation using the unit circle, we identify angles that produce specific cosine values, such as \( \cos \theta = \frac{1}{2} \) and \( \cos \theta = -\frac{1}{2} \). For these values, common angles are \( \theta = \pi/3, 5\pi/3, 2\pi/3, \) and \( 4\pi/3 \). By referencing these locations on the unit circle, we find all solutions within one full cycle of \( 0 \leq \theta < 2\pi \).

Angle Solving

Solving for angles in trigonometric equations involves identifying all possible solutions within a given interval. It requires a clear understanding of how angles on the unit circle translate to specific trigonometric values.

• First, use identities to express the equation in a simpler form.
• Second, determine the trigonometric values for which the equation holds true.
• Finally, identify all angles that produce these trigonometric values within the specified interval.

For the problem provided, the equation was simplified to \( \cos \theta = \pm \frac{1}{2} \). By consulting the unit circle, we determine that the solutions \( \theta = \pi/3, 5 \pi/3, 2\pi/3, \) and \( 4\pi/3 \) satisfy the original equation in the range \( 0 \leq \theta < 2 \pi \). This process ensures that all relevant angles are considered, capturing the periodic nature of the cosine function.