Consider the following for the next two (02) items that follow : Let f(x) = sin[π2]x + cos[-π2]x where [.] is a greatest ‘integer function
What is \(f\left(\frac{\pi}{4}\right)\) equal to ?
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 \(\frac{1}{\sqrt{2}}\).
| Expression | Value | Reason |
|---|---|---|
| \([\frac{\pi}{2}]\) | 1 | Greatest integer \(\le 1.5708\) |
| \([-\frac{\pi}{2}]\) | -2 | Greatest integer \(\le -1.5708\) |
| \(f(x)\) simplified | \(\sin(x) + \cos(2x)\) | Substitution and \(\cos(-\theta)=\cos(\theta)\) |
| \(\sin(\frac{\pi}{4})\) | \(\frac{1}{\sqrt{2}}\) | Standard trigonometric value |
| \(\cos(\frac{\pi}{2})\) | 0 | Standard trigonometric value |
| \(f(\frac{\pi}{4})\) | \(\frac{1}{\sqrt{2}}\) | Sum of \(\sin(\frac{\pi}{4})\) and \(\cos(\frac{\pi}{2})\) |
Revision Table: Key Concepts for Evaluating f(pi/4)
| Concept | Description | Application in Problem |
|---|---|---|
| Greatest Integer Function \([x]\) | Largest integer less than or equal to \(x\). | Used to simplify the coefficients of \(x\) in the \(\sin\) and \(\cos\) terms. |
| Approximation of \(\pi\) | \(\pi \approx 3.14159\) | Needed to estimate \(\frac{\pi}{2}\) and \(-\frac{\pi}{2}\) for the greatest integer calculation. |
| Trigonometric Identity | \(\cos(-\theta) = \cos(\theta)\) | Used to simplify the \(\cos[-2]x\) term. |
| Standard Angle Values | \(\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\), \(\cos(\frac{\pi}{2}) = 0\) | Essential for evaluating the function at \(x = \frac{\pi}{4}\). |
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.
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