The period of the function f(x) = |sin(x)| + |cos(x)| / |sin(x) – cos(x)| is A. Ï€/2 B. Ï€/4 C. Ï€ D. 2Ï€
The Correct Answer and Explanation is:
The given function is: f(x)=∣sin(x)∣+∣cos(x)∣∣sin(x)−cos(x)∣f(x) = \frac{| \sin(x) | + | \cos(x) |}{| \sin(x) – \cos(x) |}f(x)=∣sin(x)−cos(x)∣∣sin(x)∣+∣cos(x)∣
To find the period of this function, let’s first analyze its components. The sine and cosine functions have periods of 2π2\pi2π, but due to the absolute value and the structure of the function, the period of f(x)f(x)f(x) could be different.
Step 1: Break down the function
- ∣sin(x)∣|\sin(x)|∣sin(x)∣ and ∣cos(x)∣|\cos(x)|∣cos(x)∣ will have a period of π\piπ since both sin(x)\sin(x)sin(x) and cos(x)\cos(x)cos(x) repeat every 2π2\pi2π, but the absolute value “flips” negative values to positive, making each function repeat at half the period.
- The denominator, ∣sin(x)−cos(x)∣|\sin(x) – \cos(x)|∣sin(x)−cos(x)∣, needs further inspection. The term sin(x)−cos(x)\sin(x) – \cos(x)sin(x)−cos(x) has a period of 2π2\pi2π, but when we take the absolute value, the function will repeat every π\piπ.
Step 2: Find the combined period
Since both the numerator and the denominator involve periodic functions with period π\piπ, the overall function will repeat with the same period as the shortest period involved.
Thus, the period of the function f(x)f(x)f(x) is π\piπ.
Conclusion:
The correct answer is C. π\piπ.
