Alright, guys, let's dive into a fun little math problem where we're given two equations: x = a cos(log t) and y = a sin(log t). Our mission, should we choose to accept it, is to figure out the relationship between x and y. Sounds intriguing, right? This is a classic example of how trigonometric functions and logarithmic functions can intertwine, and it’s a neat exercise in algebraic manipulation and recognizing fundamental trigonometric identities. So, grab your thinking caps, and let's get started!

Understanding the Given Equations

First, let's break down what we have. We've got x expressed in terms of a cosine function and y expressed in terms of a sine function. Both of these functions have the same argument, which is log t. The variable a is a constant, which means it's just a number that doesn't change. This is a crucial observation because it suggests that we might be able to use trigonometric identities to eliminate the log t term and find a direct relationship between x and y. Remember, the goal here isn't just to solve for x or y individually, but to find an equation that connects them directly. Think of it like finding the equation of a circle or an ellipse, where x and y are related through a single equation. By understanding the structure of these equations, we can strategically manipulate them to reveal the hidden relationship. The presence of the constant 'a' also gives us a clue that the relationship might involve a circle, since 'a' often represents the radius in such contexts. Let's keep this in mind as we proceed with the solution!

Squaring Both Equations

The key to unlocking this problem lies in a famous trigonometric identity: sin²(θ) + cos²(θ) = 1. To make use of this identity, we need to somehow get sin² and cos² terms. A natural way to do this is by squaring both equations. So, let's square both sides of the equation x = a cos(log t). We get:

x² = a² cos²(log t)

Similarly, squaring both sides of the equation y = a sin(log t) gives us:

y² = a² sin²(log t)

Now we have x² and y² expressed in terms of cos²(log t) and sin²(log t), respectively. Notice how the constant a also gets squared, which is important for the next step. Squaring these equations might seem like a simple step, but it's a crucial one because it sets us up to use the Pythagorean trigonometric identity. Without squaring, we wouldn't be able to directly apply this identity and eliminate the trigonometric functions. This step transforms the equations into a form that allows us to combine them in a meaningful way. This is a common technique in mathematics: transforming equations into a more usable form through algebraic manipulation. Remember, the goal is to find a relationship between x and y, and squaring brings us one step closer to that goal.

Adding the Squared Equations

The next logical step is to add the two squared equations together. This is where the magic happens! Adding x² = a² cos²(log t) and y² = a² sin²(log t), we get:

x² + y² = a² cos²(log t) + a² sin²(log t)

Now, notice that we can factor out the a² from the right side of the equation:

x² + y² = a² (cos²(log t) + sin²(log t))

And here's where the trigonometric identity comes into play. We know that cos²(θ) + sin²(θ) = 1 for any angle θ. In our case, θ is log t. So, we can replace the expression in the parentheses with 1:

x² + y² = a² (1)

Which simplifies to:

x² + y² = a²

And there you have it! We've found the relationship between x and y. Adding the equations together was the pivotal move, as it allowed us to utilize the trigonometric identity and eliminate the log t term. This step demonstrates the power of algebraic manipulation and the importance of recognizing key mathematical relationships. The result is a simple and elegant equation that connects x and y directly. This is a common strategy in problem-solving: combining equations in a way that cancels out unwanted terms and reveals the underlying relationship between the variables. Let's now interpret what this equation represents.

Interpreting the Result

The equation x² + y² = a² should look familiar. It's the equation of a circle centered at the origin (0, 0) with a radius of a. So, the relationship between x and y is that they lie on a circle with radius a. This is a beautiful result that connects the seemingly complex expressions involving trigonometric and logarithmic functions to a simple geometric shape. The fact that we ended up with a circle is not entirely surprising, given the presence of sine and cosine functions, which are inherently related to circular motion and angles. This result provides a visual representation of the relationship between x and y. As t varies, the point (x, y) traces out a circle of radius a. This is a powerful example of how mathematical equations can describe geometric shapes and relationships. Moreover, it highlights the interconnectedness of different areas of mathematics, such as trigonometry, algebra, and geometry. Understanding the geometric interpretation of equations can often provide deeper insights into their meaning and behavior. In this case, the equation reveals that the relationship between x and y is fundamentally circular in nature.

Conclusion

So, to recap, given x = a cos(log t) and y = a sin(log t), we found the relationship between x and y to be x² + y² = a². This represents a circle centered at the origin with a radius of a. This problem beautifully illustrates how algebraic manipulation and trigonometric identities can be used to simplify complex expressions and reveal underlying relationships. Remember, guys, the key was to square both equations, add them together, and then use the identity sin²(θ) + cos²(θ) = 1. This allowed us to eliminate the log t term and find a direct relationship between x and y. This is a valuable technique to keep in your mathematical toolkit! Always be on the lookout for ways to use identities and algebraic manipulations to simplify equations and uncover hidden relationships. And don't forget to interpret your results geometrically whenever possible, as this can provide deeper insights and understanding. Keep practicing, and you'll become a math whiz in no time!