Problem 28 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \sec 2 x=2 $$

Step-by-Step Solution

Answer

All solutions for \( x \) are \( x = 30^\circ + 180^\circ n \) and \( x = 150^\circ + 180^\circ n \) for any integer \( n \).

Step 1: Rewrite the Equation in Terms of Cosine

Since the secant function is the reciprocal of the cosine function, we can rewrite secant as a cosine. Therefore, the equation \( \sec 2x = 2 \) becomes \( \frac{1}{\cos 2x} = 2 \).

Step 2: Solve for Cosine Function

Take the reciprocal of both sides to solve for the cosine function. This results in \( \cos 2x = \frac{1}{2} \).

Step 3: Find General Solutions for Cosine

We know that the cosine of \( 60^\circ \) (or \( \frac{\pi}{3} \) radians) and \( 300^\circ \) (or \( \frac{5\pi}{3} \) radians) is \( \frac{1}{2} \). Therefore, \( 2x = 60^\circ + 360^\circ n \) and \( 2x = 300^\circ + 360^\circ n \), with \( n \) being an integer, capture all possible angles.

Step 4: Solve for x

Divide all terms in the solution of \( 2x \) by 2 to solve for \( x \). The first solution gives \( x = 30^\circ + 180^\circ n \) and the second solution gives \( x = 150^\circ + 180^\circ n \), where \( n \) is an integer.

Step 5: Verify the Solutions

Since the solutions are based on trigonometric identities and transformations, verify by putting some values of \( n \) into the solutions for \( x \) and checking if the original equation \( \sec 2x = 2 \) holds true. Due to periodicity, the expressions will satisfy the equation for all integer \( n \).

Key Concepts

Secant Function

The secant function, often denoted as \( \sec \theta \), is a fundamental trigonometric function. It is the reciprocal of the cosine function. In mathematical terms, this means that \( \sec \theta = \frac{1}{\cos \theta} \). Understanding this relationship is crucial when solving trigonometric equations involving secant.

• The secant function is undefined where the cosine function is zero. This is because division by zero is not possible.
• Throughout trigonometry, secant is less commonly used compared to sine and cosine but is just as important for its unique properties.

Knowing that secant is directly tied to cosine allows us to convert more complex equations into manageable ones that use cosine. This translation is a key step in solving trigonometric equations.

Cosine Function

The cosine function, denoted as \( \cos \theta \), is one of the primary trigonometric functions. It measures the adjacent side over the hypotenuse in a right triangle. When dealing with trigonometric equations, cosine often appears because of its relation to secant.

• Cosine ranges from -1 to 1. This bounded range can help simplify and verify solutions.
• It is a periodic function, repeating itself every \( 360^\circ \) or \( 2\pi \) radians, which makes it consistent over repeated cycles.

When given an equation such as \( \sec 2x = 2 \), recognizing that it can be rewritten using \( \cos \) is the first step to finding a solution. Here, we derive \( \cos 2x = \frac{1}{2} \) by taking the reciprocal, which aligns with known values of cosine.

General Solutions for Trigonometric Equations

Finding the general solution for a trigonometric equation involves identifying all the possible angles that satisfy the equation. For trigonometric equations like \( \cos 2x = \frac{1}{2} \), there are specific angles, such as \( 60^\circ \) and \( 300^\circ \), known for their cosine values.

• To obtain the general solution, add multiples of the full period of the trig function (360° for cosine) to each specific angle: \( 2x = 60^\circ + 360^\circ n \) and \( 2x = 300^\circ + 360^\circ n \), where \( n \) is an integer.
• This ensures that all possible solutions are captured because the function repeats.

This method expands one solution to an infinite series of angles, accommodating the periodic nature of trigonometric functions.

Periodicity in Trigonometry

Periodicity is a core concept in trigonometry, referring to how functions like sine, cosine, and secant repeat their values in regular intervals. For both sine and cosine, this period is \( 360^\circ \) or \( 2\pi \) radians.

• The periodicity allows trigonometric functions to cycle through their range of values consistently, which is why general solutions are possible.
• By utilizing these periods, solutions to equations can be expressed in terms of a single solution plus multiples of the function's period.

This inherent repetition aids in predicting and verifying all potential solutions to trigonometric equations, ensuring that none are missed. It simplifies problems by reducing them to manageable, repeating segments.