Solving Trig Problems: Finding Values With Tangent

Hey guys! Let's dive into a cool trigonometry problem. We're given a piece of information, and we need to find the value of an expression. This is a classic type of problem that shows up quite a bit in math, and it's all about using what you know about trigonometric ratios and how they relate to each other. Specifically, we're going to be working with tangent, sine, and cosine. Don't worry if it sounds a bit intimidating at first; we'll break it down step by step. By the end, you'll see that it's not as scary as it looks, and you'll have a solid understanding of how to tackle these kinds of problems. Ready to get started? Let's go!

Understanding the Problem

Okay, so the problem gives us that 5 tan θ = 4. This is our starting point, our key piece of information. Remember that tan θ is a trigonometric ratio, and it's defined as the ratio of the sine of an angle to the cosine of that angle, or sin θ / cos θ. We are asked to find the value of the expression: (5 sin θ - 3 cos θ) / (5 sin θ + 2 cos θ). This looks a bit complicated, but we can simplify it using what we know about tan θ. The goal here is to manipulate the given information and the expression we want to evaluate in such a way that we can utilize the value of tan θ. Think of it like a puzzle: We need to find the right pieces (the trigonometric identities and relationships) to solve it. It's all about strategy and using the right tools. Keep this in mind: always look for ways to relate the given information to what you need to find. Don't get discouraged if it's not immediately obvious – that's part of the fun!

So, how do we get started? First, we'll use the given equation 5 tan θ = 4 to find the value of tan θ. This is straightforward algebra. Then, we need to figure out how to use tan θ to help us find the value of the expression. Since the expression involves sin θ and cos θ, we will try to relate these to tan θ. The magic trick here is to divide both the numerator and the denominator of the expression by cos θ. This will introduce tan θ into the expression, making it much easier to solve. This is a super important step, so make sure you understand why we're doing it. Ready? Let's get into the solution!

Step-by-Step Solution

Alright, let's break this down into simple steps. First things first: we have 5 tan θ = 4. To find the value of tan θ, we simply divide both sides of the equation by 5:

tan θ = 4 / 5

Great! Now we know that tan θ = 4/5. This is a crucial piece of information, we'll need it later. Now, let's focus on the expression we need to evaluate: (5 sin θ - 3 cos θ) / (5 sin θ + 2 cos θ). As mentioned before, the trick here is to divide both the numerator and the denominator by cos θ. This will help us introduce tan θ into the equation. Here's how it looks:

(5 sin θ - 3 cos θ) / (5 sin θ + 2 cos θ) = ((5 sin θ - 3 cos θ) / cos θ) / ((5 sin θ + 2 cos θ) / cos θ)

Now, let's simplify by dividing each term by cos θ: this gives us:

(5 sin θ / cos θ - 3 cos θ / cos θ) / (5 sin θ / cos θ + 2 cos θ / cos θ)

Remember that sin θ / cos θ = tan θ and cos θ / cos θ = 1. So, we can rewrite our expression as:

(5 tan θ - 3) / (5 tan θ + 2)

See how dividing by cos θ created this change? Now, we can substitute the value of tan θ = 4/5 into the expression. This gives us:

(5 * (4/5) - 3) / (5 * (4/5) + 2)

Let's do the math. 5 multiplied by 4/5 is 4. So the expression becomes:

(4 - 3) / (4 + 2)

Which simplifies to:

1 / 6

So, the value of the expression (5 sin θ - 3 cos θ) / (5 sin θ + 2 cos θ) is 1/6! Awesome, right? We started with an expression that looked a bit complex, and through a series of clever steps, we found the solution. Let's recap the whole process one more time to make sure everything is crystal clear.

Recap and Key Takeaways

Alright guys, let's take a quick look back at what we did. We started with the equation 5 tan θ = 4, and we needed to find the value of (5 sin θ - 3 cos θ) / (5 sin θ + 2 cos θ). The first thing was to find the value of tan θ which was simply 4/5. The critical step was dividing both the numerator and denominator of the expression by cos θ, which allowed us to rewrite the expression in terms of tan θ. This is a super important trick to remember for similar problems. We then substituted the value of tan θ and simplified the expression to get our final answer of 1/6. The key takeaways from this are:

• Understanding the definitions: Knowing the definitions of trigonometric ratios like tan θ = sin θ / cos θ is essential.
• Strategic manipulation: The ability to manipulate expressions by dividing by cos θ is crucial for solving these types of problems.
• Step-by-step approach: Breaking down the problem into smaller, manageable steps makes it much easier to solve. This avoids feeling overwhelmed.

Remember, practice makes perfect! The more you work through these kinds of problems, the more comfortable and confident you'll become. Try changing the numbers in the problem and working through it again. You can also try to find similar problems online and test yourself. Keep an eye out for problems that involve different trigonometric ratios like sine, cosine, and cotangent. These problems often require similar techniques. This is a great way to solidify your understanding and boost your math skills. Keep up the great work, and you will be a trigonometry master in no time!

Further Practice and Exploration

Now that you've worked through this problem, you're probably wondering,