Problem 6 - Precalculus with Limits

Question

In Exercises 5-10, verify that the \( x \)-values are solutions of the equation. \( \sec x - 2 = 0 \) (a) \( x = \dfrac{\pi}{3} \) (b) \( x = \dfrac{5\pi}{3} \)

Step-by-Step Solution

Answer

Yes, \( x=\dfrac{\pi}{3} \) and \( x=\dfrac{5\pi}{3} \) are the solutions of the given equation.

Step 1: Substitute \( x=\dfrac{\pi}{3} \) into the equation

Substitute \( x=\dfrac{\pi}{3} \) into the equation: \( \sec(\dfrac{\pi}{3}) - 2 = 0 \). Since the secant of \( \dfrac{\pi}{3} \) is 2, the equation simplifies to \(2 - 2 = 0\). Therefore \( x=\dfrac{\pi}{3} \) is a solution to the equation.

Step 2: Substitute \( x=\dfrac{5\pi}{3} \) into the equation

Substitute \( x=\dfrac{5\pi}{3} \) in the equation: \( \sec(\dfrac{5\pi}{3}) - 2 = 0 \). Since the secant of \( \dfrac{5\pi}{3} \) is 2, the equation simplifies to \(2 - 2 = 0\). Therefore \( x=\dfrac{5\pi}{3} \) is a solution to the equation.

Key Concepts

Secant Function

The secant function, noted as \( \sec(x) \), is an important trigonometric function.

• It is defined as the reciprocal of the cosine function, so \( \sec(x) = \frac{1}{\cos(x)} \).
• The secant function can be tricky because it is undefined wherever cosine equals zero.
• This occurs at angles where \( x = \frac{\pi}{2} + n\pi \), with \( n \) being any integer.

For this exercise, we're focused on specific values of \( x \) where secant returns a value of 2. This happens when the cosine is \( \frac{1}{2} \). Break it down like this:- If \( \cos(x) = \frac{1}{2} \), then \( \sec(x) = \frac{1}{\frac{1}{2}} = 2 \).- Knowing common angle values helps us quickly identify these solutions.

Solutions Verification

Verifying solutions in trigonometric equations involves checking that the given angle values satisfy the original equation.

• The exercise involves the equation \( \sec(x) - 2 = 0 \).
• To verify a solution, substitute the proposed \( x \)-value into the equation.

Let's apply this to our given angle:- For \( x = \frac{\pi}{3} \), substitute into \( \sec(x) - 2 = 0 \), resulting in \( \sec\left(\frac{\pi}{3}\right) - 2 = 0 \). Knowing \( \sec\left(\frac{\pi}{3}\right) = 2 \), the equation satisfies: \( 2 - 2 = 0 \).- The same process applies to \( x = \frac{5\pi}{3} \), where the secant value is also 2. So, \( \sec\left(\frac{5\pi}{3}\right) - 2 = 0 \) holds true.Thus, both \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \) are valid solutions.

Angle Substitution

Angle substitution is crucial in solving trigonometric equations. It offers a method to test if a specific angle solves the equation.When working with angle substitution:

• First, understand the trigonometric function involved and the expected result.
• Substitute the angle into the expression. In our case, into the secant function.

Here’s how:- If \( x = \frac{\pi}{3} \), replacing \( x \) yields \( \sec\left(\frac{\pi}{3}\right) - 2 = 0 \).- Calculate the secant: since \( \sec\left(\frac{\pi}{3}\right) = 2 \), compute \( 2 - 2 = 0 \).- Similar substitution and computation occur for \( x = \frac{5\pi}{3} \), and you will find \( 2 - 2 = 0 \).This helps in accurately identifying which angles meet the equation's requirement.