Problem 21 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(g(x)=\frac{1}{2} \sec \pi x\)

Step-by-Step Solution

Answer

The final answer for function g(x) will depend on the specific value of x and is given by the expression \(g(x)=\frac{1}{2}(\frac{1}{\cos(\pi x)})\).

Step 1: Convert to Cosine

The secant function is the reciprocal of the cosine function. So, g(x) can be rewritten as \(g(x)=\frac{1}{2}(\frac{1}{\cos(\pi x)})\)

Step 2: Find Cosine Value

It's necessary to determine \(\cos(\pi x)\) for some specified value of x.

Step 3: Calculate g(x)

Once we've found \(\cos(\pi x)\), we can substitute it into our expression for g(x) to get the final answer. g(x) is then simply half the reciprocal of \(\cos(\pi x)\). That's the final value we are looking for.

Key Concepts

Secant Function

The secant function is a less commonly referenced trigonometric function compared to sine and cosine, but it holds great importance in mathematics. Specifically, the secant function is the reciprocal of the cosine function. This means that for any angle \(x\), the secant function, denoted as \( \sec(x) \), is defined as \( \sec(x) = \frac{1}{\cos(x)} \).

In terms of application, the secant function can help in solving various trigonometric problems, including those involving angles and triangles. It is also useful in calculus and other areas of mathematics, where reciprocals of trigonometric functions can simplify expressions.

Key points about the secant function:

• The secant is undefined at angles where the cosine is zero, such as \(\pi/2\), \(3\pi/2\), etc., because division by zero is undefined.
• The secant function can be useful for understanding properties of periodic functions and wave patterns.
• Its graph resembles a series of arcs and is unbounded, continuous where defined, along with having a vertical asymptote wherever the cosine value equals zero.

Cosine Function

The cosine function is one of the primary trigonometric functions. It is fundamental in both theoretical and applied mathematics. The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse when the angle is measured in radians or degrees.

In an equation, it can be represented as:

\[ \cos(x) = \text{adjacent}/\text{hypotenuse} \]

Features of the cosine function:

• Cosine is a periodic function with a period of \(2\pi\).
• It oscillates between -1 and 1, which makes it particularly useful in modeling wave patterns.
• It is an even function, meaning \( \cos(-x) = \cos(x) \).

When dealing with equations like \(g(x) = \frac{1}{2} \sec \pi x\), converting secant into cosine makes it easier to analyze and compute specific values, leading to a deeper understanding of the function.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions come into play when the inverse of sine, cosine, and tangent are needed. These include secant (\(\sec\)), cosecant (\(\csc\)), and cotangent (\(\cot\)).

These functions

• Secant is the reciprocal of cosine: \(\sec(x) = \frac{1}{\cos(x)}\)
• Cosecant is the reciprocal of sine: \(\csc(x) = \frac{1}{\sin(x)}\)
• Cotangent is the reciprocal of tangent: \(\cot(x) = \frac{1}{\tan(x)}\)

Reciprocal functions are particularly helpful in dealing with mathematical expressions where the trigonometric identities can simplify solving equations. They are used frequently in calculus to transform integrals and derivatives into more manageable forms.

Using reciprocal trigonometric functions in expressions allows mathematicians and scientists to explore and connect different mathematical concepts with ease. In the example of \(g(x) = \frac{1}{2}(\frac{1}{\cos(\pi x)})\), recognizing the reciprocal function allowed for straightforward simplification, helping to find the desired values efficiently.