Finding Sine Value On Rotating CD A Trigonometric Approach

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In the realm of mathematics, particularly trigonometry, understanding the relationship between circular motion and trigonometric functions is crucial. This article delves into a fascinating problem involving the rotation of a point on a CD and its graphical representation. We aim to determine the sine value of the function that describes this motion. The problem presents a scenario where a nick on the edge of a CD rotates to the coordinates (-6, 5) during a song. Our task is to find the sine value corresponding to this position. This involves understanding the connection between coordinates on a circle and trigonometric ratios. Let's explore the steps to solve this problem and gain a deeper understanding of the underlying concepts.

Understanding the Problem

The core of the problem lies in connecting the point (-6, 5) on the coordinate plane to the sine value of the angle formed by the rotation. To tackle this, we need to recall the fundamental trigonometric relationships within a unit circle. The sine of an angle in standard position (vertex at the origin, initial side along the positive x-axis) is defined as the y-coordinate of the point where the terminal side of the angle intersects the unit circle. However, in this case, we're not explicitly given the radius of the circle. This means our first step is to determine the radius using the given coordinates. The radius, often denoted as r, is the distance from the origin (0, 0) to the point (-6, 5). We can calculate this using the distance formula, which is derived from the Pythagorean theorem. Understanding this initial setup is crucial because it lays the foundation for applying trigonometric principles to find the sine value. Furthermore, recognizing that the sine value corresponds to the y-coordinate, but potentially scaled by the radius, is key to selecting the correct answer from the provided options. This section emphasizes the importance of visualizing the problem geometrically and relating it to the unit circle concept in trigonometry. It sets the stage for the subsequent steps, where we'll perform the necessary calculations to arrive at the solution.

Calculating the Radius

To determine the sine value, we first need to calculate the radius (r) of the circle. As mentioned earlier, the radius is the distance from the origin (0, 0) to the point (-6, 5). We employ the distance formula, which is given by:

r=(x2βˆ’x1)2+(y2βˆ’y1)2r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

In our case, (x₁, y₁) = (0, 0) and (xβ‚‚, yβ‚‚) = (-6, 5). Substituting these values into the formula, we get:

r=(βˆ’6βˆ’0)2+(5βˆ’0)2r = \sqrt{(-6 - 0)^2 + (5 - 0)^2}

r=(βˆ’6)2+(5)2r = \sqrt{(-6)^2 + (5)^2}

r=36+25r = \sqrt{36 + 25}

r=61r = \sqrt{61}

Therefore, the radius of the circle is √61. This value is essential because it acts as the scaling factor when relating the coordinates of the point to the sine and cosine values. In a unit circle (radius = 1), the sine and cosine are simply the y and x coordinates, respectively. However, with a radius of √61, we need to adjust the y-coordinate to obtain the sine value. This calculation of the radius is a crucial step, as it bridges the gap between the given Cartesian coordinates and the trigonometric representation of the point's position on the rotating CD. The radius, being the hypotenuse of the right triangle formed by the x and y coordinates, allows us to use trigonometric ratios correctly. Without knowing the radius, we cannot accurately determine the sine value associated with the point (-6, 5).

Determining the Sine Value

Now that we have the radius, √61, we can calculate the sine value. Recall that the sine of an angle in standard position is defined as the ratio of the y-coordinate to the radius:

sin(ΞΈ)=yrsin(ΞΈ) = \frac{y}{r}

In this case, the y-coordinate is 5 and the radius is √61. Substituting these values, we get:

sin(ΞΈ)=561sin(ΞΈ) = \frac{5}{\sqrt{61}}

However, to match the answer choices, we need to rationalize the denominator. This means eliminating the square root from the denominator. We do this by multiplying both the numerator and denominator by √61:

sin(ΞΈ)=561βˆ—6161sin(ΞΈ) = \frac{5}{\sqrt{61}} * \frac{\sqrt{61}}{\sqrt{61}}

sin(ΞΈ)=56161sin(ΞΈ) = \frac{5\sqrt{61}}{61}

Therefore, the sine value of the angle is (5√61)/61. This result represents the vertical component of the point's position on the rotating CD, scaled by the radius. The process of rationalizing the denominator is a standard practice in mathematics to express answers in a simplified form. It ensures that the denominator is a rational number, making the expression easier to work with. The final sine value, (5√61)/61, accurately reflects the trigonometric relationship between the point (-6, 5) and the angle formed by its rotation from the origin. This calculation demonstrates the direct application of trigonometric definitions and algebraic manipulation to solve the problem.

Selecting the Correct Answer

Comparing our calculated sine value, (5√61)/61, with the provided options, we find that it matches option A:

A. 56161\frac{5 \sqrt{61}}{61}

Therefore, the correct answer is A. This step involves a straightforward comparison between the derived result and the given choices. It ensures that we have accurately followed the problem's instructions and selected the option that corresponds to our calculation. The other options can be ruled out because they either have incorrect signs, incorrect values, or fail to rationalize the denominator. Option B, - rac{6 \sqrt{61}}{61}, has the wrong sign and uses the x-coordinate instead of the y-coordinate. Option C, -√61, represents the negative of the radius and is not a sine value. Option D, √5, does not relate to the calculated ratio of the y-coordinate to the radius. The process of elimination can be a helpful strategy in multiple-choice questions, allowing us to narrow down the possibilities and confirm the correctness of our answer. Selecting the correct answer is the final step in the problem-solving process, demonstrating a clear understanding of the concepts and calculations involved.

Conclusion

In summary, we successfully determined the sine value of the function representing the rotation of a nick on a CD. By understanding the relationship between coordinates on a circle and trigonometric ratios, we calculated the radius using the distance formula and then applied the definition of sine to find the value. The sine value, (5√61)/61, corresponds to option A. This problem highlights the importance of connecting geometric concepts with trigonometric functions. The ability to visualize the problem, apply the distance formula, and use trigonometric definitions are crucial skills in mathematics. Furthermore, the process of rationalizing the denominator is a standard algebraic technique that ensures answers are expressed in a simplified form. This exercise not only reinforces our understanding of trigonometry but also demonstrates how mathematical concepts can be applied to real-world scenarios, such as the rotation of a CD. The combination of geometric intuition and algebraic manipulation is essential for solving a wide range of mathematical problems, making this a valuable learning experience.