The Maximum Value of Sin 2x: A Comprehensive Guide
Understanding the maximum value of sin 2x and its underlying principles is crucial for anyone dealing with trigonometric functions. In this guide, we will delve into the maximum value of sin 2x and explore why it is 1, which occurs at specific intervals of x.
Introduction to Sin 2x
The function sin 2x is a transformation of the basic sine function, where the input is doubled. This transformation changes the period of the sine wave, making it oscillate twice as fast. Among the real numbers, it's fascinating to explore the range of values that sin 2x can take and identify its maximum and minimum values.
Maximum Value of Sin 2x
The sine function, sin x, has a range of [-1, 1]. The maximum value of sin x is 1, and this occurs when x equals any odd multiple of π/2. For the function sin 2x, the maximum value remains 1, due to the transformation applied to x. This means that sin 2x can also reach a maximum value of 1, but at half the period of the basic sine function.
When Does Sin 2x Reach Its Maximum?
To better understand when sin 2x reaches its maximum value, we can set 2x to π/2 plus any integer multiple of 2π (i.e., 2x π/2 2kπ, where k is an integer). This gives us:
[text{x} frac{pi}{4} kpi]
For any integer k, the value of x will satisfy these conditions and hence sin 2x will reach its maximum value of 1. This explains why the maximum value of sin 2x is 1 and it occurs at these specific x-values.
Graphical Representation
Visually, the function sin 2x has a period of π. This means it completes one full cycle in π units, which is half the period of the standard sine function. The peaks and troughs of the function sin 2x will still be at the same relative points as in the standard sine function, but the frequency is doubled.
Additional Insights
The function sin 2x is closely related to the standard sine function, but the relationship can be explored mathematically through its derivative and second derivative:
Derivative Analysis
By taking the derivative of sin 2x, we find:
[f(x) sin 2x]
[f'(x) 2cos 2x 0]
[2x frac{pi}{2} kpi]
[x frac{pi}{4} frac{kpi}{2}]
Setting the first derivative to zero helps us find the critical points. At these points, the function might have local maxima or minima. However, using the second derivative to confirm:
[f''(x) -4sin 2x]
[f''left(frac{-pi}{4}right) -4sin left(-frac{pi}{2}right) 4 > 0]
This confirms that sin 2x has a local minimum at x -π/4, and similarly, sin 2x has a local maximum at x π/4, where its value is 1.
Conclusion
To summarize, the maximum value of sin 2x is 1, and it occurs at x π/4 kπ, where k is any integer. This understanding is crucial for various applications in mathematics, physics, and engineering, where trigonometric functions play a pivotal role.
Frequently Asked Questions (FAQs)
Question 1: How does the transformation from sin x to sin 2x affect the range and period?
Answer: The transformation from sin x to sin 2x doubles the frequency, making the period π instead of 2π. This means that sin 2x completes one full cycle in π units instead of 2π units. The range of both sin x and sin 2x is [-1, 1], but sin 2x oscillates twice as fast.
Question 2: Why does sin 2x attain its minimum value at x -π/4?
Answer: At x -π/4, the value of 2x is -π/2. The sine function reaches its minimum value of -1 at -π/2. This is confirmed by the second derivative test, where f''(x) -4sin 2x. Plugging in x -π/4, we get f''(-π/4) 4, which is positive, indicating a local minimum.
Question 3: Can sin 2x take any value other than -1, 0, and 1?
Answer: No, sin 2x cannot take any value outside the range [-1, 1]. The sine function, whether in its basic form or transformed as sin 2x, is bounded by -1 and 1 for all real values of its argument.