Finding Sine: Angle In Standard Position Explained

Hey math enthusiasts! Let's dive into a classic trigonometry problem. This one deals with angles in standard position and figuring out the sine of an angle. Specifically, we're given a point that the terminal side of an angle passes through and need to determine the sine value. Ready to break it down? Let's go!

Understanding the Basics: Standard Position and the Unit Circle

Alright, before we jump into the problem, let's make sure we're all on the same page with the fundamentals. When we talk about an angle in standard position, we're referring to an angle placed on the coordinate plane where:

• Its vertex (the point where the two rays meet) is at the origin (0, 0).
• Its initial side (one of the rays) lies along the positive x-axis.

Think of it like this: imagine a rotating ray that starts on the positive x-axis and swings around the origin. The angle is formed by the amount this ray rotates. Now, the terminal side is the position of that rotating ray after it's done swinging. That's the key to our problem here. This concept is best visualized in conjunction with the unit circle. The unit circle is a circle centered at the origin with a radius of 1. It's super handy in trigonometry because it provides a visual representation of the sine, cosine, and tangent values for any angle.

When the terminal side of an angle intersects the unit circle, the x-coordinate of that point of intersection is equal to the cosine of the angle, and the y-coordinate is equal to the sine of the angle. Therefore, when looking at a point $(x, y)$ that lies on the terminal side of an angle in standard position (but not necessarily on the unit circle), we need to consider the distance from the origin to that point, often referred to as 'r'.

In our case, we're not dealing with the unit circle, so we will not have a radius of 1. If we are given a point that lies on the terminal side, we can still determine the sine and cosine, but we need a little help from the Pythagorean theorem, which we will discuss next. This is like having a map of the angle's position. The point guides us, and the math helps us find the actual trig values. So, the question is really about finding the sine value using the information given, and we have to use the point on the terminal side to make it happen. Got it?

The Significance of 'r' in Trigonometry

We know that the radius of the unit circle is 1, and on the unit circle, the x and y coordinates are cosθ and sinθ, respectively. But what if we aren't dealing with a unit circle? That's where 'r' comes in. Think of 'r' as the radius of a circle that passes through your given point (x, y). We can find 'r' using the distance formula, which is essentially the Pythagorean theorem in disguise: $r = \sqrt{x^2 + y^2}$.

So, if we have a point $(x, y)$ and we know the radius 'r', we can define the trigonometric functions for any angle: $sin(\theta) = \frac{y}{r}$, $cos(\theta) = \frac{x}{r}$ and $tan(\theta) = \frac{y}{x}$.

This is all super handy because even if our point isn't on the unit circle, we can still figure out the sine, cosine, and tangent. Essentially, 'r' scales the values relative to the unit circle. It's all about proportions. The core relationship here is that the trigonometric functions, $sin(\theta)$, $cos(\theta)$, and $tan(\theta)$, are defined by ratios involving the sides of a right triangle. This is the heart of the matter when solving these types of problems. With this background, let's start solving the main problem.

Solving the Problem Step-by-Step

Now, let's put our knowledge to the test. We're given that the terminal side of an angle in standard position passes through the point P(15, -8). Our goal is to find the value of $\sin \theta$. Here’s how we break it down:

1. Identify x and y: From the point P(15, -8), we know that x = 15 and y = -8.
2. Calculate r: We can find r (the distance from the origin to the point) using the distance formula/Pythagorean theorem: $r = \sqrt{x^2 + y^2}$.
   • Substitute the values: $r = \sqrt{(15)^2 + (-8)^2}$.
   • Simplify: $r = \sqrt{225 + 64}$.
   • Calculate: $r = \sqrt{289} = 17$.
3. Find $\sin \theta$: Remember, $\sin \theta = \frac{y}{r}$.
   • Substitute the values: $\sin \theta = \frac{-8}{17}$.

Therefore, $\sin \theta = -\frac{8}{17}$.

So, the answer is B: $\sin \theta = -\frac{8}{17}$. See? Not so bad, right? We just needed a few simple steps, a good understanding of what those terms mean, and we're golden!

Visualize the Solution

It's always helpful to visualize what's going on. Imagine a right triangle formed by dropping a perpendicular line from the point (15, -8) to the x-axis. The sides of the triangle are: the horizontal side (x) with a length of 15, the vertical side (y) with a length of 8 (ignoring the negative sign for now, as it just indicates the direction), and the hypotenuse (r) with a length of 17. The angle $\theta$ is formed at the origin. Since the y-value is negative, this angle is in the fourth quadrant.

In the fourth quadrant, the sine function is negative, which aligns with our answer. This visualization helps confirm our calculations and gives us a better grasp of the geometric interpretation of the problem. This geometrical representation provides a deeper understanding of how these concepts all fit together. Thinking geometrically can make these problems much easier to visualize and solve.

Key Takeaways and Tips for Similar Problems

• Always start by identifying x, y, and calculating r using $r = \sqrt{x^2 + y^2}$.
• Remember the definitions: $\sin \theta = \frac{y}{r}$, $\cos \theta = \frac{x}{r}$, and $\tan \theta = \frac{y}{x}$. Knowing these definitions is fundamental to solving problems in trigonometry.
• Pay attention to the quadrant: The quadrant in which the terminal side of the angle lies determines the signs of the trigonometric functions. Sine is negative in quadrants III and IV, cosine is negative in quadrants II and III, and tangent is negative in quadrants II and IV.
• Practice, practice, practice! The more problems you solve, the more comfortable you'll become with these concepts. Work through various examples, and don't hesitate to draw diagrams to aid your understanding. This is how you will start to see the patterns in these problems.

Quick Tips for Exam Success

• Memorize the unit circle or at least be familiar with its key values. This will help you quickly determine the sine and cosine of common angles.
• Understand the relationships between sine, cosine, and tangent. For instance, $\tan \theta = \frac{\sin \theta}{\cos \theta}$.
• Use the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$. This is a powerful tool for solving trigonometric problems.

By following these steps and keeping these tips in mind, you'll be well-equipped to tackle similar problems and ace your trigonometry exams! Keep practicing, and you'll find these concepts become second nature.

Conclusion: Mastering Trigonometry

So, there you have it! We've successfully navigated a trigonometry problem involving angles in standard position and found the sine of an angle. We've seen how to use the coordinates of a point on the terminal side and the Pythagorean theorem to calculate 'r' and then find the sine value. We also considered the quadrant to determine the sign and understood the significance of visualization. The concepts of 'x,' 'y,' and 'r,' and their relationship with the trigonometric functions, are critical in trigonometry. Remember, practice is key. Keep working through problems, and you'll master these concepts in no time! Keep up the great work, and good luck with your math adventures!