Hey guys! Today, we're diving into a fun math problem where we need to find the maximum value of a fraction, specifically (5-4m)/(5-4n), given the condition that m^2 + n^2 = 1. This type of problem often pops up in geometry and trigonometry, and it’s super cool to see how different mathematical concepts connect. The original problem was actually about maximizing the ratio of two sides in a geometric figure, which boiled down to this neat algebraic expression. So, let's break it down step by step and figure out how to tackle it. Think of this as a mathematical adventure where we're the explorers, uncovering hidden treasures of knowledge! We'll use a mix of algebra and perhaps even a touch of trigonometry to solve this. Remember, math isn't just about formulas; it's about the journey of problem-solving. So, grab your thinking caps, and let's get started!
Understanding the Problem
So, before we jump into calculations, let’s really understand what the problem is asking. We've got this equation, m^2 + n^2 = 1. What does that remind you of? If you're thinking of the unit circle, you're on the right track! This equation represents a circle with a radius of 1 centered at the origin in the m-n plane. That means any point (m, n) that satisfies this equation lies on this circle. Now, we want to maximize the expression (5 - 4m) / (5 - 4n). This looks like a ratio, and we need to find the largest possible value this ratio can take, considering that m and n are constrained by the equation of the circle. To maximize this fraction, we either want to make the numerator as large as possible or the denominator as small as possible (or a combination of both!). This is where things get interesting. We can't just pick any values for m and n; they have to play nice with the m^2 + n^2 = 1 condition. This constraint is key to solving the problem. We need a way to relate m and n within this circle to the expression we want to maximize. We might think about using trigonometric substitutions since circles and trigonometry are good buddies. Or, we could explore algebraic manipulations to see if we can rewrite the expression in a more manageable form. The main thing is to visualize what's happening. We're essentially walking around the unit circle and checking the value of our expression at each point. The highest value we find is our answer. Let's get to it and find that maximum!
Trigonometric Substitution
Okay, let's try a clever trick using trigonometry. Since m^2 + n^2 = 1, we can think of m and n as the cosine and sine of some angle, let's call it θ (theta). So, we can say m = cos(θ) and n = sin(θ). This substitution is super handy because it automatically satisfies the m^2 + n^2 = 1 condition, thanks to the trigonometric identity cos^2(θ) + sin^2(θ) = 1. Now, let’s plug these into our expression:
(5 - 4m) / (5 - 4n) = (5 - 4cos(θ)) / (5 - 4sin(θ))
See how we've transformed the problem? Instead of dealing with m and n directly, we now have a function of a single variable, θ. This makes things a whole lot easier to analyze. Our goal now is to find the maximum value of this expression as θ varies. This is where our calculus instincts might kick in. We could try to find the critical points by taking the derivative with respect to θ and setting it equal to zero. But, before we dive into derivatives, let's take a step back and think. Is there another way? Sometimes, a bit of algebraic manipulation can save us from a messy calculation. We might be able to rewrite this expression in a form that's easier to maximize. For example, we could try to express it in terms of a single trigonometric function or use some trigonometric identities to simplify it. The key here is to be flexible and explore different approaches. Math is like a puzzle, and sometimes you need to try a few different pieces before you find the right fit. So, let's keep this trigonometric form in mind and see if we can simplify it further. Maybe we can find a clever way to make this expression shine!
Calculus Approach
Alright, let's roll up our sleeves and tackle this using calculus. We've got our expression in terms of theta: (5 - 4cos(θ)) / (5 - 4sin(θ)). To find the maximum value, we need to find the critical points. Remember, critical points are where the derivative is either zero or undefined. So, first up, let's find the derivative of our expression with respect to θ. This might look a bit intimidating, but we can handle it using the quotient rule. The quotient rule states that the derivative of (u/v) is (u'v - uv') / v^2, where u' and v' are the derivatives of u and v, respectively.
Let's identify our u and v:
- u = 5 - 4cos(θ)
- v = 5 - 4sin(θ)
Now, let's find their derivatives:
- u' = 4sin(θ)
- v' = -4cos(θ)
Plug these into the quotient rule formula, and we get:
d/dθ [(5 - 4cos(θ)) / (5 - 4sin(θ))] = [(4sin(θ) * (5 - 4sin(θ)) - (5 - 4cos(θ)) * (-4cos(θ))] / (5 - 4sin(θ))^2
Phew! That looks like a mouthful. Let's simplify this a bit. Expanding and combining like terms, we get:
[20sin(θ) - 16sin^2(θ) + 20cos(θ) - 16cos^2(θ)] / (5 - 4sin(θ))^2
We can simplify further by using the identity sin^2(θ) + cos^2(θ) = 1:
[20sin(θ) + 20cos(θ) - 16] / (5 - 4sin(θ))^2
Now, to find the critical points, we set this derivative equal to zero. A fraction is zero when its numerator is zero, so we need to solve:
20sin(θ) + 20cos(θ) - 16 = 0
This looks like a trigonometric equation we can solve! We're getting closer to finding the maximum value. Next, we'll work on solving this equation for θ. Stay tuned, guys, we're on the right track!
Solving the Trigonometric Equation
Okay, we've arrived at the trigonometric equation: 20sin(θ) + 20cos(θ) - 16 = 0. Let’s simplify this a bit by dividing everything by 4:
5sin(θ) + 5cos(θ) - 4 = 0
Now, this looks more manageable. To solve this, we can use a neat trick: let's try to rewrite the left side in the form of Rcos(θ - α) or Rsin(θ + α), where R is a constant and α is an angle. This is a common technique for solving equations involving both sine and cosine. To do this, we can rewrite the equation as:
5sin(θ) + 5cos(θ) = 4
Now, let's divide both sides by √(5^2 + 5^2) = 5√2:
(1/√2)sin(θ) + (1/√2)cos(θ) = 4 / (5√2)
Notice that 1/√2 is both the sine and cosine of π/4 (45 degrees). So, we can rewrite the left side using the sine addition formula:
sin(θ)cos(π/4) + cos(θ)sin(π/4) = sin(θ + π/4)
Our equation now becomes:
sin(θ + π/4) = 4 / (5√2)
This is much easier to deal with! Let's find the value of θ + π/4:
θ + π/4 = arcsin(4 / (5√2))
Now we can find θ:
θ = arcsin(4 / (5√2)) - π/4
But hold on a second! Remember, arcsin only gives us one solution in the range [-π/2, π/2]. There might be another solution in the interval [0, 2π). To find the other solution, we can use the fact that sin(π - x) = sin(x). So, another possible solution for θ + π/4 is:
π - arcsin(4 / (5√2))
And the corresponding θ is:
θ = π - arcsin(4 / (5√2)) - π/4
We've found two possible values for θ. Now, we need to plug these values back into our original expression (5 - 4cos(θ)) / (5 - 4sin(θ)) to see which one gives us the maximum value. This might involve some calculator work, but we're in the home stretch now!
Evaluating for Maximum Value
Great job, guys! We've found two possible values for θ:
- θ₁ = arcsin(4 / (5√2)) - π/4
- θ₂ = π - arcsin(4 / (5√2)) - π/4
Now, it's time to plug these values back into our original expression (5 - 4cos(θ)) / (5 - 4sin(θ)) and see which one gives us the maximum value. This is where we'll likely need a calculator to get accurate results. Let's start with θ₁:
(5 - 4cos(θ₁)) / (5 - 4sin(θ₁))
We can calculate the values of cos(θ₁) and sin(θ₁) using our calculator. Remember to make sure your calculator is in radian mode since we're working with radians! Once we have those values, we can plug them into the expression and get a numerical result. Now, let's do the same for θ₂:
(5 - 4cos(θ₂)) / (5 - 4sin(θ₂))
Again, we'll use our calculator to find cos(θ₂) and sin(θ₂) and then plug them into the expression. Now, we'll have two numerical values. The larger of the two is the maximum value of our expression. We've done it! We've successfully navigated through the trigonometric substitution, the calculus steps, and the trigonometric equation solving. All that's left is to compare the two values and declare our champion: the maximum value of (5 - 4m) / (5 - 4n), given m^2 + n^2 = 1. Remember, math is a journey, and we've just completed a pretty awesome one!
Conclusion
So, after all the hard work, we've reached the finish line! We used a combination of trigonometric substitution, calculus, and algebraic manipulation to find the maximum value of (5 - 4m) / (5 - 4n), given the constraint m^2 + n^2 = 1. The key steps involved recognizing the circular constraint, using trigonometric substitution to simplify the expression, employing calculus to find critical points, and solving the resulting trigonometric equation. Finally, we evaluated the original expression at the critical points to determine the maximum value. This problem is a great example of how different areas of mathematics connect and how a creative approach can lead to a solution. We started with an algebraic expression, linked it to a geometric concept (the unit circle), used trigonometric identities to simplify, applied calculus to optimize, and then solved a trigonometric equation. It’s a beautiful blend of mathematical techniques! Remember, problem-solving in mathematics isn't just about finding the right answer; it's about the process. It's about exploring different avenues, trying different techniques, and learning from the challenges along the way. This problem has given us a fantastic opportunity to practice these skills. I hope you guys enjoyed this mathematical journey as much as I did. Keep exploring, keep questioning, and keep solving! Math is an adventure, and there's always something new to discover.