Problem 58 - Calculus

Question

Give an example of: A sine function with period 23

Step-by-Step Solution

Answer

Use \( f(x) = \sin\left(\frac{2\pi}{23}x\right) \) as the sine function with period 23.

Step 1: Understanding the Sine Function

The standard sine function is given by \( f(x) = \sin(x) \). The period of the function \( \sin(x) \) is 2\( \pi \). We need to adjust this function so that it has a period of 23.

Step 2: Applying Period Formula

To change the period of a sine function, use the formula \( f(x) = \sin(bx) \), where the period is \( \frac{2\pi}{b} \). We want the period to be 23, so we set \( \frac{2\pi}{b} = 23 \).

Step 3: Solving for b

Now solve the equation \( \frac{2\pi}{b} = 23 \) for \( b \). Multiply both sides by \( b \) and divide by 23 to get \( b = \frac{2\pi}{23} \).

Step 4: Constructing the Sine Function

Substitute \( b = \frac{2\pi}{23} \) back into the function \( f(x) = \sin(bx) \). The function becomes \( f(x) = \sin\left(\frac{2\pi}{23}x\right) \). This sine function has a period of 23.

Key Concepts

Period of a Function

In mathematics, the period of a function is a key concept, especially when studying waves and oscillations. The period of a function refers to the length of one complete cycle of the repeating pattern. A function is said to be periodic if it repeats its pattern at regular intervals.
For instance, in the context of the sine function, the standard function has a period of \(2\pi\), which means it takes \(2\pi\) radians for the wave to begin repeating its cycle.
The formula for finding the period of a sine function, or any periodic function, is crucial when modifying the function's parameters to achieve a desired period.

• The formula for the period of a sine function is given by \(\frac{2\pi}{b}\), where \(b\) is a constant that modifies how stretched or compressed the cycle is.
• By changing \(b\), you can adjust the period to suit specific needs, such as setting it to 23, as in the problem statement above.

Understanding this concept allows for tremendous flexibility and application in real-world scenarios, such as modeling sound waves, light waves, and other sinusoidal patterns.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are foundational in trigonometry, a branch of mathematics dealing with angles and periodic phenomena. The most common trigonometric functions include the sine, cosine, and tangent functions.
The sine function, in particular, is vital for describing oscillatory motion such as waves. The standard sine function is expressed as \(\sin(x)\). Here, \(x\) represents the angle in radians, and its output ranges between -1 and 1.

• The sine function graph is a smooth, wave-like curve that repeats every \(2\pi\) radians.
• This sinusoidal pattern is often used to model periodic phenomena.

Trigonometric functions are not limited to theoretical mathematics. They are used in engineering, physics, and even in music theory to model sound waves. Applying the sine function with specific periods allows for the modeling and analysis of a variety of cyclic behaviors.

Frequency in Trigonometry

In trigonometry, frequency is closely related to the concept of the period. While the period tells us how long it takes for a function to complete one cycle, frequency tells us how many cycles occur in a given unit of time.
Understanding the difference between period and frequency is essential for interpreting oscillations and waves in different contexts. The relationship between period \(T\) and frequency \(f\) is inversely proportional, given by the equation \(f = \frac{1}{T}\).

• For example, if a sine wave has a period of 23, its frequency would be \(\frac{1}{23}\).
• Hence, only approximately 0.043 cycles occur in a single unit of time for this specific example.

By understanding frequency, one can discern how "fast" something oscillates. This becomes increasingly relevant in fields such as telecommunications, where signal frequencies are crucial. Frequency also plays an important role in music and acoustics, affecting the pitch and harmonics of sound.