Solving For Coordinates On Terminal Rays A Trigonometry Problem

Hey guys! Today, we're diving into a fun math problem that combines trigonometry and coordinate geometry. We're going to explore terminal rays, cosecant, and how they help us pinpoint a point on a coordinate plane. So, buckle up, and let's get started!

The Problem: Decoding the Point P(x, y)

Let's break down the problem we're tackling. We have a point P(x, y) situated on the terminal ray of an angle θ. Now, this angle θ isn't just any angle; it's nestled between π radians and 3π/2 radians. Think of it like this: we're cruising in the third quadrant of our coordinate plane. To add a twist, we're given that csc θ = -5/2. Our mission, should we choose to accept it, is to figure out the exact coordinates of point P(x, y).

Breaking Down the Basics: Terminal Rays and Angles

Before we jump into solving, let's refresh our understanding of some key concepts. Imagine a ray, like a beam of light, starting from the origin (0, 0) of our coordinate plane. This is our initial ray. Now, picture this ray rotating counterclockwise. The ray's final position after rotation is what we call the terminal ray. The angle formed between the initial ray and the terminal ray is our angle θ. Understanding this visual is crucial because the quadrant in which the terminal ray lies gives us clues about the signs of our trigonometric functions.

In this specific problem, θ lies between π and 3π/2 radians. This places our terminal ray squarely in the third quadrant. Remember, the third quadrant is where both x and y coordinates are negative. Keep this in mind, it's a golden nugget of information for later!

Cosecant: The Reciprocal Connection

Now, let's talk about cosecant (csc). Cosecant is one of our six trigonometric functions, and it's closely related to sine (sin). In fact, cosecant is the reciprocal of sine. Mathematically, we write this as csc θ = 1/sin θ. Think of it as flipping the fraction! This reciprocal relationship is super important because it allows us to switch between cosecant and sine, depending on what information we have.

We're given that csc θ = -5/2. Using our reciprocal relationship, we can find sin θ simply by flipping the fraction. So, sin θ = -2/5. Notice the negative sign? That's our third quadrant knowledge kicking in! Sine corresponds to the y-coordinate, and in the third quadrant, y-coordinates are negative. Everything's aligning perfectly!

Connecting the Dots: Sine, Coordinates, and the Unit Circle

To truly understand how sine relates to our coordinates, let's bring in the concept of the unit circle. The unit circle is a circle with a radius of 1 centered at the origin. When we draw an angle θ in standard position (initial ray along the positive x-axis), the point where the terminal ray intersects the unit circle gives us crucial information. The x-coordinate of this intersection point is equal to cos θ (cosine of θ), and the y-coordinate is equal to sin θ (sine of θ).

Now, here's the magic: we know sin θ = -2/5. This means the y-coordinate of the point where our terminal ray intersects the unit circle is -2/5. But our point P(x, y) isn't necessarily on the unit circle. It's somewhere along the terminal ray. So, we need a way to scale our unit circle values to match the actual coordinates of P(x, y).

Building the Right Triangle: Visualizing the Relationship

Let's visualize this with a right triangle. Imagine dropping a perpendicular line from point P(x, y) to the x-axis. This creates a right triangle, with the terminal ray as the hypotenuse. The length of the side opposite to angle θ is the absolute value of the y-coordinate (|y|), and the length of the side adjacent to θ is the absolute value of the x-coordinate (|x|). The hypotenuse is the distance from the origin to point P(x, y), which we can call r.

Now, remember our definition of sine: sin θ = opposite/hypotenuse. In our triangle, this translates to sin θ = y/r. We know sin θ = -2/5. This gives us a ratio between y and r. We can think of this as y = -2k and r = 5k, where k is a scaling factor. This is where the magic happens – we are finding a ratio to make our triangle similar to a triangle within the unit circle.

Finding the Missing Piece: Cosine and the Pythagorean Theorem

To find the x-coordinate, we need to involve cosine. Remember, cos θ = adjacent/hypotenuse, which in our triangle is cos θ = x/r. But we don't know x or cos θ yet! This is where the Pythagorean Theorem comes to our rescue. In our right triangle, the Pythagorean Theorem states: x² + y² = r².

We know y = -2k and r = 5k. Let's plug these values into the Pythagorean Theorem:

x² + (-2k)² = (5k)²

x² + 4k² = 25k²

x² = 21k²

x = ±√(21k²) = ±k√21

Now, remember we're in the third quadrant, where x-coordinates are negative. So, we choose the negative root: x = -k√21. We can now say that the cosine is cos θ = x/r = (-k√21)/5k = -√21/5

The Grand Finale: Determining the Coordinates of P(x, y)

We're almost there! We've established the ratios for x, y, and r in terms of k:

• x = -k√21
• y = -2k
• r = 5k

Notice that we do not need to solve for k to answer this question because we were not given a value for the radius r. All we need to do is to identify the x and y values that are in the same ratio that we just calculated. Looking at the answer options, the values P(-√21, -2) perfectly match this ratio when k=1. If we use other values for k, the results would be P(-2√21, -4) and so on.

Therefore, the coordinates of point P(x, y) are P(-√21, -2). We did it! We successfully navigated through terminal rays, cosecant, and the Pythagorean Theorem to pinpoint the location of our mystery point.

Key Takeaways and Why This Matters

This problem wasn't just about finding coordinates; it was about connecting fundamental trigonometric concepts. We saw how the quadrant of an angle influences the signs of trigonometric functions, how cosecant relates to sine, and how the unit circle helps us visualize these relationships. Understanding these connections is crucial for tackling more complex problems in trigonometry and calculus.

In summary, we:

• Refreshed our understanding of terminal rays and angles.
• Utilized the reciprocal relationship between cosecant and sine.
• Connected sine to coordinates using the unit circle concept.
• Employed the Pythagorean Theorem to find the missing coordinate.
• Considered quadrant signs to determine the correct solution.

So, the next time you encounter a problem involving trigonometric functions and coordinate points, remember these steps. Break down the problem, visualize the relationships, and don't be afraid to use the tools in your math arsenal. You've got this!

Breaking Down the Problem Keywords for Clarity

To make sure we're all on the same page, let's clarify the key elements of the problem. Sometimes, the way a question is worded can be a little confusing, so we're going to rephrase the core question to make it super clear.

Original Problem Element: "The point P(x, y) is on the terminal ray of angle θ. If θ is between π radians and 3π/2 radians and csc θ = -5/2, what are the coordinates of P(x, y)?"

Rephrased Question: "Given that point P(x, y) lies on the terminal ray of an angle θ in the third quadrant (between π and 3π/2 radians), and the cosecant of θ is -5/2, what are the x and y coordinates of point P?"

What Changed and Why:

• We explicitly stated that the angle θ is in the third quadrant. This removes any ambiguity and reinforces a critical piece of information.
• We replaced "θ is between π radians and 3π/2 radians" with "in the third quadrant (between π and 3π/2 radians)." This is more conversational and directly links the radian range to its quadrant.
• We rephrased "what are the coordinates of P(x, y)?" to "what are the x and y coordinates of point P?" This is a minor change, but it emphasizes that we're looking for both the x and y values.

By rephrasing the question, we've made it more direct and easier to understand. This is a helpful strategy whenever you're tackling math problems – break down the question into its core components and reword it in a way that makes sense to you.

Optimizing for Search: Title and Keywords

Now, let's talk about making sure this article is easily discoverable by others who might be struggling with similar problems. That means crafting a title and incorporating keywords that people are likely to search for.

Why is this important? Think of search engines like Google as vast libraries. To help people find the information they need, we need to label our article in a way that makes sense to the library's cataloging system. That's where SEO (Search Engine Optimization) comes in.

Crafting an SEO-Friendly Title

The title is the first thing people see, both in search results and on the article page itself. It needs to be clear, concise, and accurately reflect the content. It should also include relevant keywords.

Original Title: The point P(x, y) is on the terminal ray of angle θ. If θ is between π radians and 3π/2 radians and csc θ = -5/2, what are the coordinates of P(x, y) ?

SEO-Optimized Title: Finding Coordinates on Terminal Rays: Cosecant and Trigonometry Problem

What Changed and Why:

• We made it more concise. The original title was a direct restatement of the problem, which is too long for a title.
• We used action-oriented language. "Finding Coordinates" is more engaging than "The point P(x, y) is on the terminal ray..."
• We included relevant keywords. "Terminal Rays," "Cosecant," and "Trigonometry Problem" are terms people might search for when looking for help with this type of problem.

Strategic Keyword Integration

Keywords are the words and phrases that people type into search engines. To make our article visible, we need to include these keywords naturally within the content. Let's look at some key terms and how we've incorporated them:

• Terminal Ray: This phrase is central to the problem, so we've used it throughout the article, especially when explaining the concept.
• Cosecant: We've used "cosecant" and its abbreviation "csc" when discussing the trigonometric function and its relationship to sine.
• Coordinates: Since the goal is to find the coordinates of a point, we've used this term frequently in the explanation and solution.
• Trigonometry: This is a broader keyword that encompasses the overall topic, so we've included it in the title and introduction.
• Unit Circle: This concept is crucial for understanding the relationship between trigonometric functions and coordinates, so we've explained it in detail.
• Pythagorean Theorem: This is a key tool for solving the problem, so we've made sure to mention it and explain its application.

Best Practices for Keywords:

• Use keywords naturally. Don't stuff them into the text in a way that sounds forced or unnatural.
• Focus on providing valuable content. The best way to rank well in search results is to create high-quality content that people find helpful.
• Use variations of keywords. For example, instead of just using "terminal ray," we also used "terminal rays."

By optimizing our title and incorporating relevant keywords, we've increased the chances that this article will reach the people who need it most. It's all about making our content discoverable and providing valuable information to our readers.

I hope this comprehensive breakdown helps you understand the problem and the concepts behind it! Keep practicing, and you'll become a trigonometry whiz in no time!