Question

\(g(x)=4 \sec \pi x\)

Short answer

The short answer would be the computed value of \(g(x) = 4 / \cos(\pi x)\) for the specific value of \(x\). It's important to note that if \(\cos(\pi x) = 0\), \(g(x)\) will be undefined because one cannot divide by zero in mathematics.

Step-by-step solution

Defining the secant function

The function \(g(x) = 4 \sec \pi x\) can be rewritten by replacing the secant function \(\sec\) with its definition as the reciprocal of the cosine function \(\cos\). This gives \(g(x) = 4 / \cos(\pi x)\). Therefore, to find the value of \(g(x)\) for any \(x\), first calculate the value of \(\cos(\pi x)\), then divide 4 by this value.

Calculate \(\cos(\pi x)\)

To find the value of \(g(x)\), determine the value of \(\cos(\pi x)\) for your specific \(x\). The value of \(\cos(\pi x)\) varies between -1 and 1. Use a calculator to compute this step if needed.

Compute \(g(x)\)

After finding the value of \(\cos(\pi x)\), calculate the value of \(4 / \cos(\pi x)\). This will give the value of \(g(x)\). An important point to note is that whenever \(\cos(\pi x) = 0\), \(g(x)\) will be undefined because division by zero is undefined in mathematics.

Key concepts

Secant Function

The secant function, denoted by \( \sec \), is a trigonometric function that is the reciprocal of the cosine function \( \cos \). This means that for any angle \( x \), the secant can be expressed as \( \sec(x) = \frac{1}{\cos(x)} \). The secant function is undefined wherever the cosine value is zero, as division by zero is not possible. Since cosine varies between -1 and 1, the secant function can take any value from negative infinity to positive infinity, except at the points where cosine is zero. The secant function is periodic, and it repeats every \( 2\pi \) interval. This function is particularly important in trigonometry, partly because it appears in many complex trigonometric identities and transformations.

Cosine Function

The cosine function, represented as \( \cos \), is a fundamental trigonometric function that describes the ratio of the adjacent side to the hypotenuse in a right triangle. Cosine values range from -1 to 1 as the angle \( x \) varies. It is known for its wavelike graph called a cosine wave, which begins at 1, decreases to -1, and returns to 1 over an interval of \( 2\pi \). The cosine function is even, which means \( \cos(-x) = \cos(x) \), and this property contributes to its symmetry around the y-axis. Understanding the behavior of the cosine function is crucial as it forms the basis for understanding other trigonometric functions, including the secant function.

Reciprocal Functions

Reciprocal functions arise when you take one divided by a function. In trigonometry, the secant, cosecant, and cotangent functions are the reciprocal functions of cosine, sine, and tangent, respectively. These functions are critical for understanding the relationships between different trigonometric ratios. The reciprocal of a function means you are essentially flipping its numerator and denominator. For instance, the reciprocal of the cosine function is secant: \( \sec(x) = \frac{1}{\cos(x)} \). A key point to remember is that wherever the original function is zero, the reciprocal is undefined. This is crucial in determining the domain and range of reciprocal trigonometric functions.

Function Evaluation

Function evaluation involves finding the specific output of a function for a given input. With trigonometric functions, it often includes determining the cosine or sine of an angle and then applying any additional operations. To evaluate the function \( g(x) = 4 \sec(\pi x) \), follow these steps:

• Compute \( \pi x \), which creates the new angle to evaluate.
• Determine \( \cos(\pi x) \) using a calculator or a known value sheet.
• Calculate \( \sec(\pi x) = \frac{1}{\cos(\pi x)} \) to find the secant.
• Multiply the result by 4 to obtain \( g(x) \).

This process shows how the evaluation of functions connects algebraic manipulations with trigonometric identifications, ensuring you obtain correct values for specific inputs.