Understanding the Function with Greatest Integer

The given function is defined as \(f(x) = \sin[\frac{\pi}{2}]x + \cos[-\frac{\pi}{2}]x\), where \([.]\) represents the greatest integer function. The greatest integer function of a real number \(y\), denoted by \([y]\), is the largest integer less than or equal to \(y\).

To evaluate \(f(x)\), we first need to determine the values of the greatest integer parts: \([\frac{\pi}{2}]\) and \([-\frac{\pi}{2}]\).

Evaluating the Greatest Integer Parts

We know that the value of \(\pi\) is approximately \(3.14159\).

Let's calculate \(\frac{\pi}{2}\):

\[ \frac{\pi}{2} \approx \frac{3.14159}{2} \approx 1.5708 \]

The greatest integer less than or equal to \(1.5708\) is \(1\). So,

\[ \left[\frac{\pi}{2}\right] = 1 \]

Now let's calculate \(-\frac{\pi}{2}\):

\[ -\frac{\pi}{2} \approx -1.5708 \]

The greatest integer less than or equal to \(-1.5708\) is \(-2\) (since \(-2 \le -1.5708\) and \(-1 > -1.5708\)). So,

\[ \left[-\frac{\pi}{2}\right] = -2 \]

Simplifying the Function Definition

Now substitute these integer values back into the function definition:

\[ f(x) = \sin(1 \cdot x) + \cos(-2 \cdot x) \]

This simplifies to:

\[ f(x) = \sin(x) + \cos(-2x) \]

Using the trigonometric identity \(\cos(-\theta) = \cos(\theta)\), we can simplify the term \(\cos(-2x)\):

\[ \cos(-2x) = \cos(2x) \]

So, the simplified function is:

\[ f(x) = \sin(x) + \cos(2x) \]

Calculating \(f\left(\frac{\pi}{4}\right)\)

We need to find the value of the function \(f(x)\) when \(x = \frac{\pi}{4}\). Substitute \(x = \frac{\pi}{4}\) into the simplified function:

\[ f\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) + \cos\left(2 \cdot \frac{\pi}{4}\right) \]

Simplify the term inside the cosine function:

\[ 2 \cdot \frac{\pi}{4} = \frac{2\pi}{4} = \frac{\pi}{2} \]

So the expression becomes:

\[ f\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{2}\right) \]

Now, we use the known values of trigonometric functions at these standard angles:

\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

\[ \cos\left(\frac{\pi}{2}\right) = 0 \]

Substitute these values into the expression for \(f\left(\frac{\pi}{4}\right)\):

\[ f\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + 0 \]

\[ f\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

Thus, \(f\left(\frac{\pi}{4}\right)\) is equal to \(\frac{1}{\sqrt{2}}\).

Revision Table: Key Concepts for Evaluating f(pi/4)

Additional Information: Properties of Greatest Integer Function and Trigonometry

The greatest integer function, also known as the floor function, has the property that \(x - 1 < [x] \le x\). It is a step function that is constant between consecutive integers and jumps at integer values.

Understanding trigonometric identities like \(\cos(-\theta) = \cos(\theta)\) and \(\sin(-\theta) = -\sin(\theta)\) is crucial for simplifying expressions. Also, knowing the values of trigonometric functions for common angles such as \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi\) is fundamental for solving many problems involving trigonometric functions.

The problem combines concepts from both number theory (greatest integer function) and trigonometry, requiring careful evaluation of each part.