How to find the period of y = sin(Bx) in Pre-Calculus.

📚 Understanding the Period of $y = \sin(Bx)$

The period of a trigonometric function like sine represents the length of one complete cycle of the wave. For the standard sine function, $y = \sin(x)$, the period is $2\pi$. However, when we introduce a coefficient 'B' inside the sine function, as in $y = \sin(Bx)$, it affects the period. Let's explore how!

📜 Historical Context

Trigonometric functions have been studied for centuries, with roots tracing back to ancient astronomy and geometry. Understanding their periodic nature was crucial for predicting celestial events and mapping the world. The concept of a function's period became formalized with the development of calculus and more rigorous mathematical analysis.

🔑 Key Principles for Finding the Period

• 🔍 The Standard Sine Function: The period of $y = \sin(x)$ is $2\pi$. This means the sine wave completes one full cycle from 0 to $2\pi$.
• 🔄 Impact of the Coefficient 'B': In the function $y = \sin(Bx)$, the coefficient 'B' compresses or stretches the sine wave horizontally. A larger 'B' compresses the wave, resulting in a shorter period, while a smaller 'B' stretches it, leading to a longer period.
• 📐 Formula for the Period: The period of $y = \sin(Bx)$ is given by the formula: $\text{Period} = \frac{2\pi}{|B|}$. The absolute value of B ensures the period is always positive.
• ➕ Understanding Frequency: 'B' essentially determines the frequency of the wave. Frequency is the number of cycles completed per unit of $x$.

📝 Step-by-Step Example

Let's find the period of $y = \sin(3x)$:

1. Identify $B$: Here, $B = 3$.
2. Apply the Formula: $\text{Period} = \frac{2\pi}{|3|} = \frac{2\pi}{3}$.

Therefore, the period of $y = \sin(3x)$ is $\frac{2\pi}{3}$.

🌍 Real-World Examples

• 📡 Radio Waves: The sine function models electromagnetic waves, including radio waves. The coefficient 'B' affects the frequency of the radio wave.
• 🎶 Sound Waves: Sound waves can also be modeled using sine functions. Different values of 'B' will change the pitch and tone.
• 💡 Electrical Circuits: Alternating current (AC) in electrical circuits follows a sinusoidal pattern. The frequency of AC power (e.g., 60 Hz in the US) is directly related to the 'B' value in the sine function.

🧪 Practice Quiz

Find the period of each sine function:

1. $y = \sin(2x)$
2. $y = \sin(\frac{1}{2}x)$
3. $y = \sin(-4x)$
4. $y = 3\sin(5x)$
5. $y = -2\sin(\frac{2}{3}x)$
6. $y = \sin(\pi x)$
7. $y = 5\sin(\frac{\pi}{2}x)$

Answers:

1. $\pi$
2. $4\pi$
3. $\frac{\pi}{2}$
4. $\frac{2\pi}{5}$
5. $3\pi$
6. $2$
7. $4$

✅ Conclusion

Understanding how to find the period of $y = \sin(Bx)$ is fundamental in pre-calculus. By recognizing the impact of the coefficient 'B' and applying the formula $\frac{2\pi}{|B|}$, you can easily determine the period of any sine function in this form. This knowledge is applicable across various fields, from physics and engineering to music and electronics.