Problem 47 - Algebra and Trigonometry

Question

\(43-52\) a Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi) .\) \(\cos 2 \theta-\cos ^{2} \theta=0\)

Step-by-Step Solution

Answer

The solutions are \( \theta = 0 \) and \( \theta = \pi \).

Step 1: Use the double angle identity

The double angle identity for cosine is \( \cos 2\theta = 2\cos^2\theta - 1 \). Substitute this identity into the equation \( \cos 2\theta - \cos^2\theta = 0 \). This gives us \( 2\cos^2\theta - 1 - \cos^2\theta = 0 \).

Step 2: Simplify the equation

Simplify the equation by combining like terms. \( 2\cos^2\theta - \cos^2\theta \) becomes \( \cos^2\theta \), so our equation becomes \( \cos^2\theta - 1 = 0 \).

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

Set \( \cos^2\theta - 1 = 0 \) equal to zero and solve for \( \cos^2\theta \). By adding 1 to both sides, we get \( \cos^2\theta = 1 \). Take the square root of both sides, which gives \( \cos\theta = \pm 1 \).

Step 4: Find angles \( \theta \) in interval \([0, 2\pi)\)

The values of \( \theta \) for which \( \cos\theta = 1 \) are \( \theta = 0 \) and for which \( \cos\theta = -1 \) is \( \theta = \pi \). Therefore, the solutions in the given interval \([0, 2\pi)\) are \( \theta = 0 \) and \( \theta = \pi \).

Key Concepts

Double Angle Formulas

In trigonometry, double angle formulas are crucial identities used to simplify expressions and solve equations involving trigonometric functions. These formulas express trigonometric functions of angles that are double the usual angle.

The double angle formula for cosine is particularly useful and is given by:

• \[ \cos 2\theta = 2\cos^2\theta - 1 \]

This formula helps to transform and simplify equations, allowing us to find solutions that might not be immediately obvious.

When applying these formulas, you'll often need to substitute the identity directly into the equation you are solving. Using the double angle formula can lead to expressions that are easier to manipulate. In our case, substituting into \( \cos 2 \theta - \cos^2 \theta = 0 \) helped in condensing the equation further.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, representing the x-coordinate of a point on the unit circle as the angle varies.

Key characteristics of the cosine function include:

• It varies between -1 and 1.
• It is periodic with a period of \(2\pi\).
• For certain angles such as 0 and \(\pi\), the cosine takes on values of \(1\) and \(-1\) respectively.

In solving equations like \( \cos^2 \theta = 1 \), it important to understand these properties.

The cosine function allows quick identification of angles that satisfy the equation. For instance, the angles at which \( \cos \theta = \pm 1 \) are easily identified using these properties. So in the equation, \( \theta = 0 \) and \( \theta = \pi \) satisfy the cosine values found.

Trigonometric Equations

Trigonometric equations involve trigonometric functions and are solved within a given domain. Solving these equations usually involves using trigonometric identities, such as the double angle formula, to reduce and simplify the expressions.

Here’s how to tackle trigonometric equations effectively:

• Identify if any known trigonometric identities can be applied, such as double angle or Pythagorean identities.
• Substitute these identities into the equation to simplify.
• Solve for the trigonometric function, such as \(\cos \theta\) or \(\sin \theta\).
• Determine the specific solutions within the given interval.

For example, in our case: - Simplify \( \cos 2 \theta - \cos^2 \theta = 0 \) using a double angle identity.
- Solve the resulting equation\( \cos^2 \theta = 1 \) to find \( \cos \theta = \pm 1 \).- Determine where this condition holds within \([0, 2\pi)\), leading to solutions \(\theta = 0\) and \(\theta = \pi\).