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 \(f\left(\frac{\pi}{2}\right)\) equal to ?
0
Understanding the Function and the Greatest Integer Concept
The given function is \(f(x) = \sin\left(\left[\frac{\pi}{2}\right]x\right) + \cos\left(\left[-\frac{\pi}{2}\right]x\right)\), where \([.]\) denotes the greatest integer function. The greatest integer function \([y]\) gives the largest integer less than or equal to \(y\).
Evaluating the Greatest Integer Expressions
First, let's evaluate the terms inside the greatest integer functions:
- \(\frac{\pi}{2}\) is approximately \(\frac{3.14159}{2} \approx 1.5708\).
- \(\left[\frac{\pi}{2}\right] = [1.5708]\). The greatest integer less than or equal to 1.5708 is 1. So, \(\left[\frac{\pi}{2}\right] = 1\).
Next:
- \(-\frac{\pi}{2}\) is approximately \(-\frac{3.14159}{2} \approx -1.5708\).
- \(\left[-\frac{\pi}{2}\right] = [-1.5708]\). The greatest integer less than or equal to -1.5708 is -2. So, \(\left[-\frac{\pi}{2}\right] = -2\).
Simplifying the Function f(x)
Now, substitute these integer values back into the function definition:
\(f(x) = \sin(1 \cdot x) + \cos(-2 \cdot x)\)
\(f(x) = \sin(x) + \cos(-2x)\)
Recall the trigonometric identity \(\cos(-\theta) = \cos(\theta)\). Applying this identity:
\(f(x) = \sin(x) + \cos(2x)\)
This is the simplified form of the function \(f(x)\).
Evaluating f(\(\frac{\pi}{2}\))
We need to find the value of the function at \(x = \frac{\pi}{2}\). Substitute \(x = \frac{\pi}{2}\) into the simplified function:
\(f\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) + \cos\left(2 \cdot \frac{\pi}{2}\right)\)
\(f\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) + \cos(\pi)\)
Using Standard Trigonometric Values
Now, we use the known values of sine and cosine for the angles \(\frac{\pi}{2}\) and \(\pi\):
- \(\sin\left(\frac{\pi}{2}\right) = 1\)
- \(\cos(\pi) = -1\)
Substitute these values into the expression for \(f\left(\frac{\pi}{2}\right)\):
\(f\left(\frac{\pi}{2}\right) = 1 + (-1)\)
\(f\left(\frac{\pi}{2}\right) = 1 - 1\)
\(f\left(\frac{\pi}{2}\right) = 0\)
Conclusion
The value of \(f\left(\frac{\pi}{2}\right)\) is 0.
| Step | Calculation | Result |
|---|---|---|
| 1 | Evaluate \(\left[\frac{\pi}{2}\right]\) | 1 |
| 2 | Evaluate \(\left[-\frac{\pi}{2}\right]\) | -2 |
| 3 | Substitute into \(f(x)\) | \(f(x) = \sin(x) + \cos(-2x)\) |
| 4 | Simplify \(f(x)\) using \(\cos(-\theta)=\cos(\theta)\) | \(f(x) = \sin(x) + \cos(2x)\) |
| 5 | Substitute \(x=\frac{\pi}{2}\) into \(f(x)\) | \(f\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) + \cos(\pi)\) |
| 6 | Substitute known trigonometric values | \(f\left(\frac{\pi}{2}\right) = 1 + (-1)\) |
| 7 | Final Calculation | \(f\left(\frac{\pi}{2}\right) = 0\) |
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| Greatest Integer Function \([x]\) | The largest integer less than or equal to \(x\). For example, \([3.7] = 3\), \([-2.1] = -3\), \([5] = 5\). |
| \(\sin\left(\frac{\pi}{2}\right)\) | The sine of 90 degrees or \(\frac{\pi}{2}\) radians, which is 1. |
| \(\cos(\pi)\) | The cosine of 180 degrees or \(\pi\) radians, which is -1. |
| \(\cos(-\theta)\) Identity | The cosine function is an even function, meaning \(\cos(-\theta) = \cos(\theta)\) for any angle \(\theta\). |
Additional Information: Trigonometric Functions and Radians
Trigonometric functions like sine and cosine relate angles of a right triangle to ratios of its sides, but they can also be defined for any real number using the unit circle. In calculus and many other areas of mathematics, angles are typically measured in radians rather than degrees.
- One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.
- The relationship between degrees and radians is \(180^\circ = \pi\) radians.
- So, \(90^\circ = \frac{\pi}{2}\) radians and \(180^\circ = \pi\) radians.
Understanding these fundamental angle measures and the properties of trigonometric functions is crucial for solving problems involving trigonometric expressions.
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