Problem 48 - Precalculus with Limits

Question

In Exercises 45-48, find the \( x \)-intercepts of the graph. \( y = \sec^4 \left(\dfrac{\pi x}{8} \right) - 4 \)

Step-by-Step Solution

Answer

The x-intercepts of the graph are \( x = \dfrac{8}{3}, \dfrac{16}{3} \).

Step 1: Set y to be 0

Since we are looking for x-intercepts, we need to set \( y = 0 \) and solve for \( x \). Therefore, the equation will be \( 0 = \sec^4 \left(\dfrac{\pi x}{8} \right) - 4 \)

Step 2: Solve for secant

Move 4 to the other side of the equation, making the equation \( \sec^4 \left(\dfrac{\pi x}{8} \right) = 4 \). Then, we take the fourth root on both sides, obtaining \( \sec \left(\dfrac{\pi x}{8} \right) = \pm 2 \)

Step 3: Solve for x

Use the definition of secant as the inverse of cosine, so \( \cos \left(\dfrac{\pi x}{8} \right) = \pm \dfrac{1}{2} \). This means \( \dfrac{\pi x}{8} = \arccos \left(\pm \dfrac{1}{2} \right) \), which is \( \dfrac{\pi x}{8} = \dfrac{\pi}{3}, \dfrac{2\pi}{3} \). Get the values of \( x \) by multiplying both sides by 8, obtaining \( x = \dfrac{8}{3}, \dfrac{16}{3} \).

Key Concepts

Secant Function

The secant function, denoted as secant or sec, is one of the six fundamental trigonometric functions. Unlike the more familiar sine and cosine functions, the secant function is not as commonly mentioned in basic trigonometry, but it is equally important. It is defined as the reciprocal of the cosine function, which means that for any angle \( \theta \), the secant of \( \theta \) is given by \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

The graph of the secant function exhibits a series of U-shaped curves called lobes, occurring wherever the cosine function does not equal zero. The function grows without bound as it approaches the vertical lines \( \theta = \frac{(2n + 1)\pi}{2} \), where \( n \) is any integer, as these are the points where the cosine function equals zero and the secant function is undefined.

Characteristics of Secant Function:

• It is periodic with a period of \( 2\pi \) radians.
• It has vertical asymptotes where the cosine function is zero.
• It can take on any real value, from negative to positive infinity.
• It is an even function, meaning that \( \sec(-\theta) = \sec(\theta) \).

Understanding the secant function is crucial for solving various trigonometric equations where the secant function or its reciprocal, cosine, are present.

Solving Trigonometric Equations

Solving trigonometric equations is a common task in mathematics that involves finding all the angles that make the equation true. These equations can range from simple expressions involving a single trigonometric function to more complex ones that require the use of identities and inverse functions.

To solve a trigonometric equation:

1. Isolate the trigonometric function, if possible.
2. Determine the general solution, which often involves using trigonometric identities or algebraic manipulation.
3. Take into account the specific domain or interval over which you are solving, whether it's \( 0 \) to \( 2\pi \), all real numbers, or a different interval.
4. Make use of inverse trigonometric functions, if necessary, to determine the angle(s) in question.

When faced with more complex functions, like \( \sec^4(x) \), it can be helpful to reduce the equation to a simpler one using algebraic transformations and then solve for the basic trigonometric function involved, in this case, the secant function (or its cosine counterpart).

Remember that trigonometric equations can have multiple solutions due to the periodic nature of trigonometric functions. Therefore, it's essential to be thorough in finding all possible solutions within the given domain.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the standard trigonometric functions, and they allow us to find angles when given trigonometric function values. Each of the six trigonometric functions—sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)—has an inverse. These are often denoted with an \( -1 \) superscript, such as \( \arcsin \), \( \arccos \), and \( \arctan \).

These functions are particularly useful when solving trigonometric equations. For example, if \( \cos(\theta) = \frac{1}{2} \), to find \( \theta \), you might use the inverse cosine function: \( \theta = \arccos(\frac{1}{2}) \).

Key Points to Remember:

• The range of values for \( \arcsin \) and \( \arccos \) is \( -\pi/2 \) to \( \pi/2 \) and \( 0 \) to \( \pi \), respectively.
• These functions return principal values, which are the angles within their primary range that correspond to the trigonometric function values.
• Inverse trigonometric functions can be graphed and have their own sets of properties and uses.

Understanding and employing inverse trigonometric functions is essential for extracting angle measures from the given trigonometric function value, thus providing a foundational tool for solving trigonometric equations.