Finding The Period Of Y = (1/2)sin(6θ - (3π/4)) + 3 - A Step-by-Step Guide

Hey there, math enthusiasts! Ever wondered how to decipher the period of a sinusoidal function? Today, we're diving deep into the fascinating world of trigonometry to unravel the mystery behind the function $y = \frac{1}{2} \sin(6\theta - \frac{3\pi}{4}) + 3$. This comprehensive guide will not only provide you with the answer but also equip you with the knowledge to tackle similar problems with confidence. So, grab your calculators and let's get started!

Understanding Sinusoidal Functions and Periodicity

Before we jump into the specifics, let's lay the groundwork by understanding the fundamental concepts of sinusoidal functions and their periodicity. Sinusoidal functions, like sine and cosine, are the backbone of many natural phenomena, from sound waves to the oscillations of a pendulum. Their periodic nature means they repeat their values at regular intervals. The period of a sinusoidal function is the length of one complete cycle, essentially the distance along the x-axis (or in our case, the ${\theta}$-axis) it takes for the function to repeat its pattern.

Think of it like a rollercoaster. The period is the length of the track from the starting point to the point where the ride begins to repeat its ups and downs. For the basic sine function, $y = \sin(x)$, the period is $2\pi$. This means the sine wave completes one full cycle between 0 and $2\pi$. But what happens when we start tweaking the equation, adding coefficients, and shifting things around? That's where the real fun begins!

The general form of a sinusoidal function is given by:

$y = A \sin(B(x - C)) + D$

Where:

• A is the amplitude, which determines the vertical stretch of the function.
• B affects the period of the function.
• C is the horizontal shift (phase shift).
• D is the vertical shift.

The period of the function is calculated using the formula:

$\text{Period} = \frac{2\pi}{|B|}$

This formula is our key to unlocking the period of any sinusoidal function. Let's keep this in mind as we dissect our given equation.

Dissecting the Given Equation: $y = \frac{1}{2} \sin(6\theta - \frac{3\pi}{4}) + 3$

Now, let's turn our attention to the specific equation we're tasked with: $y = \frac{1}{2} \sin(6\theta - \frac{3\pi}{4}) + 3$. Our mission is to identify the value of 'B' in this equation, as it's the key to calculating the period. The equation might look a bit intimidating at first, but let's break it down step by step.

First, notice that the equation is in a slightly modified form compared to the general form we discussed earlier. We have a term inside the sine function that needs to be factored to clearly identify 'B'. The term is $6\theta - \frac{3\pi}{4}$. To make it look more like $B(x - C)$, we need to factor out the coefficient of ${\theta}$, which is 6. So, let's rewrite the equation:

$y = \frac{1}{2} \sin\left(6\left(\theta - \frac{\pi}{8}\right)\right) + 3$

Ah, much better! Now we can clearly see that $B = 6$. This is the value that directly influences the period of our function. The other components, like the amplitude (${\frac{1}{2}}$), the phase shift (${\frac{\pi}{8}}$), and the vertical shift (3), affect the shape and position of the graph but not its period.

Calculating the Period: Applying the Formula

With the value of 'B' securely in our grasp, we can now calculate the period using the formula we introduced earlier:

$\text{Period} = \frac{2\pi}{|B|}$

In our case, $B = 6$, so we plug it into the formula:

$\text{Period} = \frac{2\pi}{|6|} = \frac{2\pi}{6} = \frac{\pi}{3}$

Voila! The period of the function $y = \frac{1}{2} \sin(6\theta - \frac{3\pi}{4}) + 3$ is ${\frac{\pi}{3}}$. This means the function completes one full cycle over an interval of ${\frac{\pi}{3}}$ units along the ${\theta}$-axis. This result corresponds to option C from the choices provided.

So, what does this period actually mean in the context of the graph? Imagine the sine wave oscillating up and down. The period ${\frac{\pi}{3}}$ tells us that after every ${\frac{\pi}{3}}$ units on the ${\theta}$-axis, the wave will repeat its pattern. A smaller period means the wave is compressed horizontally, oscillating more rapidly, while a larger period means the wave is stretched out, oscillating more slowly.

Why the Other Options Are Incorrect: A Quick Elimination Round

Let's briefly discuss why the other options are incorrect. This will solidify our understanding and help us avoid common pitfalls in future problems.

• A. $14\pi$: This value is significantly larger than what we calculated. It's unlikely that the period would be such a large multiple of ${\pi}$ given the coefficient of ${\theta}$ in the equation.
• B. $16\pi$: Similar to option A, this value is far too large and doesn't align with the principles of sinusoidal function periods.
• D. $\frac{\pi}{4}$: This value is closer to the correct answer, but it's still incorrect. The period is determined by dividing $2\pi$ by the absolute value of 'B', which in our case is 6, not 8.

By understanding why these options are incorrect, we reinforce our grasp of the underlying concepts and improve our problem-solving skills.

Mastering Sinusoidal Functions: Tips and Tricks

Now that we've successfully navigated this problem, let's arm ourselves with some tips and tricks to master sinusoidal functions in general. These strategies will come in handy when you encounter similar problems in your mathematical journey.

1. Always identify the general form: Recognizing the general form of a sinusoidal function ($y = A \sin(B(x - C)) + D$ or $y = A \cos(B(x - C)) + D$) is the first step. It allows you to quickly pinpoint the key parameters.
2. Factor out 'B': If the equation isn't in the standard factored form, make sure to factor out the coefficient of the variable (in our case, ${\theta}$) to correctly identify 'B'.
3. Use the period formula: The formula $\text{Period} = \frac{2\pi}{|B|}$ is your best friend. Memorize it and apply it diligently.
4. Understand the impact of each parameter: Know how each parameter (A, B, C, and D) affects the graph of the function. This will give you a visual intuition for the problem.
5. Practice, practice, practice: The more problems you solve, the more comfortable you'll become with sinusoidal functions. Seek out practice problems and challenge yourself.
6. Visualize the graph: If possible, try to visualize the graph of the function. This can help you understand the concept of the period more intuitively. You can use graphing calculators or online tools to plot the function and observe its behavior.

Real-World Applications: Where Sinusoidal Functions Shine

Sinusoidal functions aren't just abstract mathematical concepts; they're powerful tools that describe a multitude of real-world phenomena. Understanding their properties, including the period, allows us to model and analyze these phenomena effectively. Let's explore some fascinating applications:

• Sound Waves: Sound travels in waves, and these waves can be modeled using sinusoidal functions. The period of the wave corresponds to the pitch of the sound – a shorter period means a higher pitch, and a longer period means a lower pitch.
• Light Waves: Light, like sound, also travels in waves. Sinusoidal functions can describe the electromagnetic waves that make up light. The period of a light wave determines its color – different periods correspond to different colors in the spectrum.
• Electrical Circuits: Alternating current (AC) in electrical circuits oscillates sinusoidally. The period of the oscillation is a crucial parameter in circuit design and analysis.
• Pendulums and Oscillations: The motion of a pendulum or any oscillating system can be approximated using sinusoidal functions. The period of the oscillation depends on the physical properties of the system.
• Tides: The rise and fall of tides are influenced by the gravitational forces of the Moon and the Sun, resulting in a periodic pattern that can be modeled using sinusoidal functions. The period of the tidal cycle is related to the lunar cycle.
• Seasons: The Earth's tilt and its orbit around the Sun cause the seasons, which exhibit a periodic pattern. Sinusoidal functions can be used to model the changes in temperature and daylight hours throughout the year.
• Biological Rhythms: Many biological processes, such as the sleep-wake cycle and hormone secretion, follow rhythmic patterns that can be modeled using sinusoidal functions. These rhythms have characteristic periods that influence our health and well-being.

By recognizing the ubiquity of sinusoidal patterns in the world around us, we gain a deeper appreciation for the power and relevance of these mathematical functions.

Conclusion: Conquering the Period and Beyond

We've reached the end of our journey, and what a journey it has been! We've successfully decoded the period of the function $y = \frac{1}{2} \sin(6\theta - \frac{3\pi}{4}) + 3$, discovered the formula for calculating the period, and explored the fascinating world of sinusoidal functions. Remember, guys, math isn't just about numbers and equations; it's about understanding the patterns and relationships that govern our world.

By mastering the concepts of periodicity and sinusoidal functions, you've armed yourself with a powerful tool for tackling a wide range of mathematical and real-world problems. So, keep practicing, keep exploring, and never stop questioning. The world of mathematics is vast and exciting, and there's always something new to discover. Keep your curiosity alive, and you'll be amazed at what you can achieve. Now, go forth and conquer those trigonometric challenges!