Finding Sin Θ The Terminal Side Of An Angle Passes Through P(15,-8)

In trigonometry, understanding the relationship between the coordinates of a point on the terminal side of an angle in standard position and the trigonometric functions is crucial. This article delves into how to find the value of ${\sin \theta}$ when the terminal side of an angle in standard position passes through a given point. We will explore the concepts, formulas, and steps involved in solving such problems, providing a comprehensive guide for students and enthusiasts alike.

Understanding Standard Position and Terminal Side

In trigonometry, an angle is said to be in standard position when its vertex is at the origin (0,0) of the coordinate plane, and its initial side lies along the positive x-axis. The terminal side is the ray that rotates from the initial side to form the angle. When the terminal side of an angle passes through a point (x, y), we can form a right triangle by drawing a perpendicular line from the point to the x-axis. The sides of this triangle will help us determine the values of the trigonometric functions.

When dealing with trigonometric functions, especially sine, cosine, and tangent, it's essential to understand their definitions in the context of a unit circle and right triangles. The unit circle is a circle with a radius of 1 centered at the origin. Any point (x, y) on the unit circle corresponds to an angle ${\theta}$ in standard position, where the terminal side of the angle intersects the circle. The coordinates x and y are directly related to the cosine and sine of the angle, respectively. Specifically, ${\cos \theta = x}$ and ${\sin \theta = y}$.

For a point not on the unit circle, we can still use the same principles by considering a right triangle formed by the point, the origin, and the x-axis. The coordinates (x, y) become the legs of the right triangle, and the distance from the origin to the point becomes the hypotenuse, often denoted as r. The trigonometric functions are then defined as ratios of the sides of this triangle. For example, ${\sin \theta}$ is the ratio of the opposite side (y) to the hypotenuse (r), ${\cos \theta}$ is the ratio of the adjacent side (x) to the hypotenuse (r), and ${\tan \theta}$ is the ratio of the opposite side (y) to the adjacent side (x).

Understanding these foundational concepts is crucial for solving problems where we need to find trigonometric function values given a point on the terminal side of an angle. The ability to visualize the angle in standard position, construct the reference triangle, and apply the definitions of the trigonometric functions will lead to accurate solutions. Moreover, familiarity with the signs of the trigonometric functions in different quadrants is essential for determining the correct sign of the result. For instance, sine is positive in the first and second quadrants and negative in the third and fourth quadrants, which can be crucial when the given point lies in one of the quadrants where sine is negative.

Determining ${r}$ (the Distance from the Origin)

When the terminal side of an angle in standard position passes through a point ${P(x, y)}$, we need to find the distance ${r}$ from the origin to the point ${P}$. This distance ${r}$ represents the hypotenuse of the right triangle formed by the x-coordinate, y-coordinate, and the line segment connecting the origin to the point ${P}$. We can find ${r}$ using the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as:

${ r^2 = x^2 + y^2 }$

To find ${r}$, we take the square root of both sides of the equation:

${ r = \sqrt{x^2 + y^2} }$

It is important to note that ${r}$ is always a positive value because it represents a distance. Given the coordinates of point ${P}$ as ${(15, -8)}$, we can calculate ${r}$ by substituting the values of ${x}$ and ${y}$ into the formula:

${ r = \sqrt{15^2 + (-8)^2} }$

${ r = \sqrt{225 + 64} }$

${ r = \sqrt{289} }$

${ r = 17 }$

Thus, the distance ${r}$ from the origin to the point ${(15, -8)}$ is 17. This value is crucial because it serves as the denominator in the definitions of sine and cosine in the context of the unit circle and trigonometric functions.

The significance of finding ${r}$ extends beyond just plugging values into a formula. It provides a physical interpretation of the relationship between the point in the coordinate plane and the angle formed in standard position. The value of ${r}$ normalizes the coordinates ${x}$ and ${y}$ in a way that allows us to relate them directly to the trigonometric functions. This understanding is essential for applications in physics, engineering, and other fields where trigonometric functions are used to model periodic phenomena and geometric relationships.

Furthermore, calculating ${r}$ is a foundational step in determining all six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) associated with the angle. Once ${r}$ is known, these functions can be easily found using the definitions based on the ratios of the sides of the right triangle. Therefore, mastering the calculation of ${r}$ is a critical skill in trigonometry, enabling a deeper understanding of trigonometric concepts and their applications.

Calculating ${\sin \theta}$

With the distance ${r}$ calculated, we can now determine the value of ${\sin \theta}$. The sine function is defined as the ratio of the y-coordinate to the distance ${r}$. Mathematically, this is expressed as:

${ \sin \theta = \frac{y}{r} }$

In our case, the point ${P}$ is ${(15, -8)}$, and we have found that ${r = 17}$. Therefore, we substitute the values of ${y}$ and ${r}$ into the formula:

${ \sin \theta = \frac{-8}{17} }$

Thus, the value of ${\sin \theta}$ is ${-\frac{8}{17}}$. This result indicates that the sine of the angle is negative, which makes sense because the point ${(15, -8)}$ lies in the fourth quadrant, where the y-coordinate is negative, and consequently, the sine function is also negative.

The interpretation of ${\sin \theta}$ as ${-\frac{8}{17}}$ provides valuable insight into the angle's position and characteristics. The magnitude of ${\sin \theta}$ (i.e., ${\frac{8}{17}}$) represents the ratio of the opposite side to the hypotenuse in the right triangle formed. The negative sign indicates that the angle terminates in either the third or fourth quadrant, where the sine function is negative. This information is crucial in various applications, such as navigation, physics, and engineering, where understanding the direction and magnitude of angles is essential.

Moreover, knowing ${\sin \theta}$ allows us to find other trigonometric functions if needed. For instance, if we wanted to find ${\cos \theta}$, we could use the x-coordinate and ${r}$, or we could use the Pythagorean identity ${\sin^2 \theta + \cos^2 \theta = 1}$. The ability to calculate ${\sin \theta}$ and other trigonometric functions from a given point on the terminal side of an angle is a fundamental skill in trigonometry and opens the door to solving more complex problems involving angles and triangles.

In summary, calculating ${\sin \theta}$ involves understanding its definition as the ratio of the y-coordinate to the distance ${r}$, applying the formula correctly, and interpreting the result in the context of the coordinate plane. This skill is not only essential for trigonometry but also for various scientific and engineering applications where angles and their trigonometric functions play a crucial role.

Identifying the Correct Option

Based on our calculations, we found that ${\sin \theta = -\frac{8}{17}}$. Now, we need to identify the option that matches this value.

Looking at the given options:

A. ${\sin \theta = -\frac{15}{17}}$ B. ${\sin \theta = -\frac{8}{17}}$ C. ${\sin \theta = \frac{8}{17}}$ D. ${\sin \theta = \frac{15}{17}}$

Option B, ${\sin \theta = -\frac{8}{17}}$, matches our calculated value. Therefore, the correct answer is option B.

This final step in the problem-solving process emphasizes the importance of careful calculation and attention to detail. It's not enough to correctly apply the formulas and methods; one must also accurately match the result with the available options. In multiple-choice questions, it's common for distractors (incorrect options) to be designed to catch common errors. For example, options C and D have the correct magnitude but the wrong sign, which might be chosen by someone who forgets that sine is negative in the fourth quadrant. Option A uses the x-coordinate instead of the y-coordinate in the sine calculation, another common mistake.

The process of elimination can also be a valuable strategy in identifying the correct option. If we had initially been unsure of our calculation, we could have eliminated options C and D because we knew that sine should be negative in the fourth quadrant. This would have narrowed our choices to A and B, making it easier to double-check our work and select the correct answer.

Moreover, this problem-solving approach reinforces the importance of understanding the underlying concepts and principles of trigonometry. By knowing the definitions of the trigonometric functions and their signs in different quadrants, we can not only solve problems but also gain a deeper appreciation for the mathematical relationships involved. This conceptual understanding is crucial for tackling more complex problems and for applying trigonometry in various real-world contexts.

In conclusion, the ability to identify the correct option after performing the calculations is a critical skill that complements the mathematical techniques. It requires careful comparison, attention to detail, and a solid understanding of the concepts involved, ensuring that the final answer is accurate and meaningful.

Conclusion

In conclusion, we have successfully determined the value of ${\sin \theta}$ when the terminal side of an angle in standard position passes through the point ${P(15, -8)}$. By understanding the concepts of standard position, terminal side, the Pythagorean theorem, and the definition of the sine function, we calculated ${r}$ and then ${\sin \theta}$. This step-by-step approach ensures accuracy and reinforces the fundamental principles of trigonometry. The correct answer is ${\sin \theta = -\frac{8}{17}}$, which corresponds to option B.

This exercise highlights the importance of a systematic approach to problem-solving in mathematics. Each step, from understanding the question to calculating ${r}$, finding ${\sin \theta}$, and identifying the correct option, builds upon the previous one. A clear understanding of the underlying concepts, such as the definitions of trigonometric functions and the relationship between points in the coordinate plane and angles in standard position, is crucial for success.

Moreover, this problem demonstrates the practical application of trigonometry in various fields. The ability to determine trigonometric function values from given points is essential in areas such as navigation, physics, engineering, and computer graphics. For instance, in navigation, angles and their trigonometric functions are used to calculate bearings and distances. In physics, they are used to analyze forces and motion. In engineering, they are used in structural design and analysis. And in computer graphics, they are used to rotate and transform objects in 3D space.

The skills and concepts covered in this article are not only valuable for academic purposes but also for real-world applications. By mastering these concepts, students and enthusiasts can develop a deeper appreciation for the power and versatility of trigonometry. Furthermore, the problem-solving strategies discussed here, such as breaking down the problem into smaller steps and double-checking the calculations, can be applied to a wide range of mathematical and scientific problems.

In summary, the process of finding ${\sin \theta}$ given a point on the terminal side of an angle in standard position is a fundamental skill in trigonometry that combines conceptual understanding with practical calculation. The solution not only provides the value of ${\sin \theta}$ but also reinforces the importance of a systematic approach to problem-solving and the wide-ranging applications of trigonometry in various fields.