Problem 26 - Algebra and Trigonometry

Question

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ \sec \frac{\theta}{2}=\cos \frac{\theta}{2} $$

Step-by-Step Solution

Answer

The solutions in [0, 2π) are θ = 0.

Step 1: Rewrite the equation

The given equation is \( \sec \frac{\theta}{2} = \cos \frac{\theta}{2} \). We know that the secant function is the reciprocal of the cosine function. Thus, we can rewrite the left side of the equation as \( \frac{1}{\cos \frac{\theta}{2}} \). The equation becomes \( \frac{1}{\cos \frac{\theta}{2}} = \cos \frac{\theta}{2} \).

Step 2: Multiply through by \( \cos \frac{\theta}{2} \)

To eliminate the fraction, multiply both sides by \( \cos \frac{\theta}{2} \), which gives \( 1 = \cos^2 \frac{\theta}{2} \).

Step 3: Solve for \( \cos \frac{\theta}{2} \)

The equation \( 1 = \cos^2 \frac{\theta}{2} \) implies that \( \cos \frac{\theta}{2} = \pm 1 \).

Step 4: Find all values of \( \frac{\theta}{2} \)

When \( \cos \frac{\theta}{2} = 1 \), it means \( \frac{\theta}{2} = 2k\pi \) where \( k \) is any integer. Similarly, when \( \cos \frac{\theta}{2} = -1 \), \( \frac{\theta}{2} = (2k+1)\pi \).

Step 5: Solve for \( \theta \)

For the case \( \frac{\theta}{2} = 2k\pi \), we have \( \theta = 4k\pi \). For \( \frac{\theta}{2} = (2k+1)\pi \), we have \( \theta = (4k+2)\pi \).

Step 6: Find solutions in the interval \([0, 2\pi)\)

Considering the interval \( [0, 2\pi) \), the solutions are determined as follows:\- For \( \theta = 4k\pi \), the valid \( \theta \) is when \( k=0 \), giving \( \theta = 0 \).\- For \( \theta = (4k+2)\pi \), using \( k=0 \), we get \( \theta = 2\pi \). Since \( 2\pi \) is not less than \( 2\pi \), it is considered at the boundary.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec \theta \), is a key concept in trigonometry. It serves as the reciprocal of the cosine function. Specifically, \( \sec \theta = \frac{1}{\cos \theta} \). This relationship helps in transforming trigonometric equations like the one in our problem.
When working with trigonometric identities and equations involving secant, it's common to replace secant functions with their equivalent cosine formulations. This simplification allows for easier calculations and the application of other trigonometric identities.
In practical terms, when using the secant function, remember:

• If \( \cos \theta = 0 \), then \( \sec \theta \) is undefined, as division by zero is not possible.
• Secant is closely related to tangent, as it can also be expressed using sine and cosine: \( \sec \theta = \frac{1}{\cos \theta} = \frac{\sin \theta}{\sin \theta} \cdot \frac{1}{\cos \theta} \).

This understanding is crucial when solving equations or analyzing graphs involving secant functions.

Cosine Function

The cosine function, denoted as \( \cos \theta \), is one of the fundamental building blocks in trigonometry. It expresses the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. This function is periodic, with a period of \( 2\pi \), which means it repeats its values over every \( 2\pi \) interval.
In the context of the exercise, the cosine function helps simplify the given equation due to its reciprocal relationship with the secant function. Rewriting \( \sec \frac{\theta}{2} = \frac{1}{\cos \frac{\theta}{2}} \) allows us to clear fractions by multiplying through by \( \cos \frac{\theta}{2} \), reducing the problem to solving a quadratic expression in cosine.
Key aspects of the cosine function include:

• The cosine of an angle achieves its maximum value of 1 when the angle is a multiple of \( 2\pi \).
• Conversely, \( \cos \theta = -1 \) at odd multiples of \( \pi \).

Understanding these properties is essential for solving trigonometric equations and identifying solutions over specified intervals.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the appearing variables where both sides are defined. These identities are indispensable tools in simplifying and solving trigonometric problems.
In the given exercise, employing trigonometric identities helps transition from the secant to cosine functions effortlessly. Using the identity \( \sec \theta = \frac{1}{\cos \theta} \) was a key first step in solving the problem.
Some of the most frequently used trigonometric identities include:

• Reciprocal identities, like \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \).
• Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Even and odd identities, demonstrating that \( \cos(-\theta) = \cos \theta \) but \( \sin(-\theta) = -\sin \theta \).

By using these identities, you can simplify complex expressions, derive new relationships, and find solutions to challenging problems with relative ease.