Rewriting Trigonometric Functions A Comprehensive Guide

Hey everyone! Today, we're diving into the fascinating world of trigonometric functions and how to rewrite them. Specifically, we're going to tackle the problem of expressing a function in the form of a single sine wave with a phase shift. This is a super useful skill in various fields, from physics to engineering, so let's get right to it! Our mission is to rewrite the function y(t) = 2sin(4πt) + 5cos(4πt) into the form y(t) = Asin(ωt + φ). To achieve this, the initial step involves determining the amplitude, denoted as A. We'll use the provided values, c₁ = A sin φ and c₂ = A cos φ, along with some trigonometric identities and algebraic manipulations. This transformation allows us to represent the function as a single sine wave, making it easier to analyze its behavior and properties. The amplitude A provides crucial information about the wave's intensity, while the phase shift φ indicates how the wave is horizontally displaced. Understanding these parameters is essential for a comprehensive grasp of the function's characteristics. This form simplifies the analysis of oscillations and wave phenomena, making it a fundamental tool in many scientific and engineering applications. The rewritten function helps in visualizing and interpreting the wave's motion, predicting its future behavior, and designing systems that interact with it. So, let's embark on this journey to unravel the mysteries of trigonometric transformations and learn how to express complex functions in simpler, more manageable forms.

Finding the Amplitude (A)

Okay, so the first thing we need to do is find the amplitude, A. This is like the volume of our sine wave, how high and low it goes. Remember those formulas they gave us? c₁ = A sin φ and c₂ = A cos φ? These are key! In our case, we can see that c₁ = 2 (the coefficient of the sine term) and c₂ = 5 (the coefficient of the cosine term). The core concept here is the Pythagorean identity, which states that sin²(φ) + cos²(φ) = 1. This identity is the backbone of our calculation because it allows us to relate the coefficients c₁ and c₂ to the amplitude A. By squaring both sides of the equations c₁ = A sin φ and c₂ = A cos φ, we get c₁² = A² sin²(φ) and c₂² = A² cos²(φ). Adding these two equations together, we have c₁² + c₂² = A² sin²(φ) + A² cos²(φ). Factoring out A² on the right side, we obtain c₁² + c₂² = A²(sin²(φ) + cos²(φ)). Now, using the Pythagorean identity, we can simplify the expression inside the parentheses to 1, resulting in c₁² + c₂² = A². This equation is the key to finding the amplitude A. Substituting the values of c₁ and c₂, we can calculate A. This method is not just a mathematical trick; it's a powerful technique rooted in the fundamental properties of trigonometric functions. It allows us to transform a sum of sine and cosine waves into a single sine wave, simplifying analysis and making it easier to understand the underlying phenomena. By squaring and adding the coefficients, we effectively capture the total magnitude of the oscillations, which is precisely what the amplitude A represents.

To find A, we're going to use a little trick from trigonometry: squaring both equations and adding them together. Why? Because it gets rid of the pesky sin φ and cos φ terms and leaves us with something we can work with. Let's do it! Squaring c₁ = A sin φ gives us c₁² = A² sin² φ. Similarly, squaring c₂ = A cos φ gives us c₂² = A² cos² φ. Now, we add these two equations: c₁² + c₂² = A² sin² φ + A² cos² φ. Notice anything cool? We can factor out A² on the right side: c₁² + c₂² = A² (sin² φ + cos² φ). And here's the magic: we all know from trig class that sin² φ + cos² φ = 1! So, our equation simplifies to c₁² + c₂² = A². Awesome! Now, we just plug in our values for c₁ and c₂: 2² + 5² = A², which is 4 + 25 = A², or 29 = A². To get A, we just take the square root of both sides: A = √29. And there we have it! The amplitude of our rewritten function is √29. Guys, this is a big step! We've figured out how "loud" our sine wave is going to be. This is super important because it tells us the maximum displacement of our function. Whether you're dealing with sound waves, electrical signals, or mechanical vibrations, the amplitude is a key indicator of the intensity or strength of the wave. The fact that we can determine this amplitude by simply using the coefficients of the sine and cosine terms is a testament to the elegance and power of trigonometric identities. This technique is not only useful in mathematics but also has practical applications in various fields of science and engineering. It's a fundamental tool for analyzing and understanding oscillatory phenomena, allowing us to make predictions and design systems that interact with these phenomena in a controlled and predictable manner. So, let's celebrate this achievement and move on to the next exciting step!

Calculating the Phase Shift (φ)

Alright, now that we've nailed down the amplitude, it's time to tackle the phase shift, which is represented by the Greek letter φ (phi). The phase shift tells us how much our sine wave is shifted to the left or right. It's like the wave's starting position on the x-axis. To find φ, we're going to use the relationships c₁ = A sin φ and c₂ = A cos φ again, but this time, we'll focus on their ratio. Dividing c₁ by c₂ gives us c₁ / c₂ = (A sin φ) / (A cos φ). The amplitudes A cancel out, leaving us with c₁ / c₂ = sin φ / cos φ. And guess what? sin φ / cos φ is just tan φ! So, we have the simple equation tan φ = c₁ / c₂. This is awesome because it directly relates the phase shift φ to the known coefficients c₁ and c₂. We can use the arctangent function (also known as the inverse tangent), denoted as arctan or tan⁻¹, to solve for φ. The arctangent function gives us the angle whose tangent is a given value. However, it's essential to be a little careful here. The arctangent function has a range of (-π/2, π/2), which means it only gives us angles in the first and fourth quadrants. Depending on the signs of c₁ and c₂, the actual phase shift φ might be in a different quadrant. To determine the correct quadrant, we need to look at the signs of sin φ and cos φ, which are directly related to the signs of c₁ and c₂, respectively. If both c₁ and c₂ are positive, φ is in the first quadrant. If c₁ is positive and c₂ is negative, φ is in the second quadrant. If both c₁ and c₂ are negative, φ is in the third quadrant. And if c₁ is negative and c₂ is positive, φ is in the fourth quadrant. This careful consideration of quadrants ensures that we get the correct phase shift, which is crucial for accurately representing the sine wave's horizontal displacement.

So, we have tan φ = c₁ / c₂ = 2 / 5. To find φ, we need to take the arctangent (or inverse tangent) of both sides: φ = arctan(2 / 5). If you plug this into your calculator, you'll get an angle, but here's a little gotcha: the arctangent function only gives you angles in a certain range. We need to be sure we're in the right quadrant! To figure this out, let’s think about our original equations: c₁ = A sin φ = 2 and c₂ = A cos φ = 5. Since both c₁ and c₂ are positive, that means both sin φ and cos φ are positive. And where are both sine and cosine positive? In the first quadrant! So, the angle our calculator gives us is indeed the correct phase shift. Using a calculator, arctan(2 / 5) ≈ 0.38 radians. So, our phase shift φ is approximately 0.38 radians. We're getting closer and closer to our final form! Understanding the phase shift is crucial because it tells us the initial position of our wave. It's like knowing where the runner starts the race – it affects the entire race! In many applications, the phase shift can have significant implications. For example, in electrical engineering, the phase shift between voltage and current can determine the power factor of a circuit. In acoustics, the phase shift between two sound waves can cause interference, either constructive or destructive. Therefore, accurately calculating and interpreting the phase shift is essential for a comprehensive understanding of wave phenomena. Now that we've successfully determined the phase shift, we're just one step away from rewriting our original function in the desired form. We've already found the amplitude and the phase shift, and we know the angular frequency. It's time to put all the pieces together and express our function as a single sine wave with a phase shift.

Putting It All Together

Okay, guys, the moment we've been waiting for! We've found the amplitude (A), which is √29, and the phase shift (φ), which is approximately 0.38 radians. There's one more thing we need to identify, and that's the angular frequency, represented by ω (omega). Looking back at our original equation, y(t) = 2sin(4πt) + 5cos(4πt), we can see that the term inside the sine and cosine functions is 4πt. This means that our angular frequency, ω, is 4π. Remember, the angular frequency tells us how fast the wave is oscillating. It's related to the frequency (f) by the equation ω = 2πf, where f is the number of cycles per second (Hertz). Now, we have all the pieces of the puzzle! We know A, φ, and ω. We can finally write our function in the form y(t) = Asin(ωt + φ). Plugging in the values we found, we get: y(t) = √29 sin(4πt + 0.38). Boom! We did it! We've successfully rewritten our original function as a single sine wave with a phase shift. This is a significant achievement because it allows us to easily analyze and understand the behavior of the function. The rewritten form provides a clear picture of the wave's amplitude, frequency, and phase shift, which are crucial parameters for understanding oscillatory phenomena. This transformation is not just a mathematical exercise; it's a powerful tool with numerous applications in science and engineering. It allows us to simplify complex signals, analyze their frequency content, and design systems that interact with them in a controlled and predictable manner. Whether you're dealing with sound waves, electrical signals, or mechanical vibrations, the ability to rewrite functions in this form is an invaluable skill. So, let's take a moment to appreciate the journey we've taken, from the initial challenge to the final solution. We've navigated through trigonometric identities, algebraic manipulations, and the nuances of the arctangent function. And now, we stand at the summit, with a clear and concise representation of our function.

Final Answer

So, the final answer is: y(t) = √29 sin(4πt + 0.38). You've successfully rewritten the function in the desired form! High fives all around! This is awesome, guys! We took a function that looked like a sum of sine and cosine waves and turned it into a single, elegant sine wave. This makes it way easier to visualize and analyze the function. We can immediately see its amplitude, its frequency, and its phase shift. This skill is invaluable in many areas, including physics, engineering, and signal processing. Think about analyzing sound waves, electrical circuits, or mechanical vibrations – being able to rewrite functions like this gives you a powerful tool for understanding and manipulating these phenomena. The key takeaways from this exercise are the importance of trigonometric identities, the power of algebraic manipulation, and the careful consideration of quadrants when using the arctangent function. These are fundamental concepts that will serve you well in your mathematical journey. And remember, guys, math isn't just about memorizing formulas and procedures; it's about understanding the underlying principles and using them creatively to solve problems. This exercise is a perfect example of how mathematical tools can be applied to transform complex expressions into simpler, more manageable forms. So, keep exploring, keep questioning, and keep pushing your mathematical boundaries. The world of mathematics is vast and fascinating, and there's always something new to discover. And with that, we conclude our journey of rewriting trigonometric functions. Congratulations on mastering this skill, and I hope this explanation has been helpful and insightful. Keep up the great work, and remember to practice these techniques to solidify your understanding. Until next time, happy mathing!