Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \sec x+1=\sec x+3$$

Step-by-Step Solution

Answer

Solutions are \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\).

Step 1: Simplify the Equation

Begin by simplifying the given equation, \[2 \sec x + 1 = \sec x + 3.\]Subtract \(\sec x\) from both sides to isolate terms involving \(\sec x\).\[2 \sec x - \sec x + 1 = 3.\]

Step 2: Isolate Secant Function

After simplification, you get:\[\sec x + 1 = 3.\]Subtract 1 from both sides:\[\sec x = 2.\]

Step 3: Solve for Cosine Function

Recall that \(\sec x = \frac{1}{\cos x}\), hence \(\cos x = \frac{1}{2}\). Now, solve for \(x\) in the interval \([0, 2\pi)\).

Step 4: Determine Solutions for \(x\)

The equation \(\cos x = \frac{1}{2}\) corresponds to angles where the cosine value is \(\frac{1}{2}\). The standard angles that satisfy this are \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\), as these are the angles where the cosine equals \(\frac{1}{2}\) within the given interval.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec x \), is one of the six fundamental trigonometric functions. It is closely related to the cosine function, representing the reciprocal of cosine. This means that:

• \( \sec x = \frac{1}{\cos x} \)

This relationship is vital because it allows us to convert problems involving secant into more manageable problems involving cosine, which is often simpler to solve.
In practical terms,

• \( \sec x \) is undefined whenever \( \cos x = 0 \).
• When \( \sec x \geq 1 \) or \( \sec x \leq -1 \), \( x \) corresponds to angles where the cosine function yields these reciprocal values.

Understanding this function's behavior is crucial for solving trigonometric equations, as seen in our original exercise, where the equation \( \sec x = 2 \) is simplified to \( \cos x = \frac{1}{2} \), making the problem easier to manage.

Cosine Function

The cosine function, symbolized as \( \cos x \), is another core trigonometric function, measuring the adjacent side over the hypotenuse in a right triangle. It's defined on the interval \([0, 2\pi)\) when considering one full rotation around a circle.
For many trigonometric equations, the cosine function provides principal values that identify specific angles where certain conditions are met.

• At \( \cos x = \frac{1}{2} \), the angles within \([0, 2\pi)\) are \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \). These are key angles typically derived from the unit circle.

The practice of transforming an equation from a secant to a cosine, such as \( \sec x = 2 \) to \( \cos x = \frac{1}{2} \), is common because solving for cosine is straightforward due to its well-known values at specific angles.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. It is commonly used to indicate the domain where solutions to an equation are sought.
In trigonometry, we often solve for angles within a specific range, such as \([0, 2\pi)\), to ensure solutions fall within typical periods for functions like sine and cosine.

• Square brackets \([ \text{or} ]\) indicate that an endpoint is included in the interval (closed interval).
• Parentheses \(( \text{or} )\) signify that an endpoint is not included (open interval).

For instance, in the exercise, the interval \([0, 2\pi)\) specifies that we are looking for solutions from 0 up to but not including \(2\pi\), covering one complete revolution in the positive direction.