Finding Sine Theta Terminal Side Passes Through P(15 -8)

by Admin 57 views

In the realm of trigonometry, understanding the relationship between angles and their trigonometric functions is crucial. One fundamental concept is the terminal side of an angle in standard position and how it relates to the sine, cosine, and tangent of the angle. This article delves into this concept, focusing on how to determine the value of $\sin \theta$ when the terminal side of an angle in standard position passes through a given point. We will explore the necessary steps, underlying principles, and provide a detailed explanation to solve the problem, ultimately enhancing your grasp of trigonometry.

To truly grasp the concept of finding the value of $\sin \theta$, we must first lay a solid foundation in the basics of trigonometry and the standard position of an angle. Trigonometry, at its core, is the study of the relationships between the sides and angles of triangles. The primary trigonometric functions – sine (sin), cosine (cos), and tangent (tan) – are defined based on these relationships within a right-angled triangle. When we talk about angles in standard position, we're referring to angles placed on the Cartesian coordinate plane with their initial side along the positive x-axis and their vertex at the origin. The terminal side is the ray that rotates from the initial side to form the angle. The coordinates of a point on this terminal side, along with the distance from the origin to that point, provide the key to unlocking the values of the trigonometric functions. Understanding these fundamentals is paramount, as they form the building blocks for more advanced concepts in trigonometry and calculus. So, let's embark on this journey, ensuring we have a firm grasp on these foundational principles before we delve into the intricacies of solving the problem at hand.

The terminal side of an angle in standard position passes through the point P(15, -8). What is the value of $\sin \theta$?

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

To determine the value of $\sin \theta$, we need to follow a series of logical steps. These steps involve understanding the relationship between the point on the terminal side, the distance from the origin, and the trigonometric functions. By carefully applying the definitions of these functions, we can arrive at the correct solution.

  1. Visualize the Angle: Imagine an angle $ heta$ in standard position on the Cartesian plane. The initial side of the angle lies along the positive x-axis, and the vertex is at the origin (0, 0). The terminal side of this angle passes through the point P(15, -8). This means that if you were to draw a line segment from the origin to the point (15, -8), that line segment would represent the terminal side of the angle. The fact that the y-coordinate is negative tells us that the angle terminates in the fourth quadrant. Visualizing the angle in this way helps to understand the signs of the trigonometric functions, as sine is negative in the third and fourth quadrants.

  2. Determine the Coordinates (x, y): The point P(15, -8) gives us the x-coordinate and y-coordinate directly. In this case, x = 15 and y = -8. These coordinates are crucial because they relate to the adjacent and opposite sides of a right triangle formed by dropping a perpendicular from the point P to the x-axis. The x-coordinate corresponds to the adjacent side, and the y-coordinate corresponds to the opposite side. The sign of the y-coordinate is especially important as it dictates the sign of the sine function. Since y is negative, we expect the sine of the angle to be negative as well. This is a key step in ensuring we select the correct answer from the given options.

  3. Calculate the Distance (r): The distance 'r' from the origin (0, 0) to the point P(15, -8) is calculated using the Pythagorean theorem. This distance represents the hypotenuse of the right triangle we visualized earlier. The formula for calculating r is:

    r=x2+y2r = \sqrt{x^2 + y^2}

    Substituting the values x = 15 and y = -8, we get:

    r=152+(βˆ’8)2=225+64=289=17r = \sqrt{15^2 + (-8)^2} = \sqrt{225 + 64} = \sqrt{289} = 17

    So, the distance r is 17. This value is essential because it forms the denominator in the definitions of both sine and cosine functions. A common mistake is forgetting to take the square root, so it's vital to double-check this step. The distance r is always positive, as it represents a physical length.

  4. Apply the Definition of Sine: The sine of an angle in standard position is defined as the ratio of the y-coordinate to the distance r. Mathematically, this is expressed as:

    sin⁑θ=yr\sin \theta = \frac{y}{r}

    We have already determined that y = -8 and r = 17. Substituting these values into the formula, we get:

    sin⁑θ=βˆ’817=βˆ’817\sin \theta = \frac{-8}{17} = -\frac{8}{17}

    Therefore, the value of $\sin \theta$ is -8/17. This result aligns with our earlier observation that the sine function should be negative in the fourth quadrant. This step is the culmination of the previous steps, bringing together the coordinates and the distance to calculate the sine of the angle. It’s crucial to remember the definition of sine in terms of y and r to arrive at the correct answer.

  5. Select the Correct Option: Comparing our calculated value of $\sin \theta = -\frac{8}{17}$ with the given options, we find that it matches option B. Therefore, the correct answer is:

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

In conclusion, determining the value of $\sin \theta$ when the terminal side of an angle in standard position passes through a given point involves a clear understanding of trigonometric principles and a systematic approach. By visualizing the angle, identifying the coordinates, calculating the distance from the origin, and applying the definition of sine, we can accurately find the value of $\sin \theta$. This process reinforces the fundamental relationship between angles and their trigonometric functions, which is crucial for further studies in trigonometry and related fields. The problem-solving strategy outlined here not only provides a solution to the specific question but also equips you with a methodology that can be applied to a wide range of trigonometric problems. Remember, practice is key to mastering these concepts, so make sure to apply this knowledge to various examples to solidify your understanding.

To summarize, the key steps in solving this type of problem are: first, visualize the angle in standard position on the Cartesian plane. This provides a geometric understanding of the problem and helps anticipate the sign of the trigonometric function. Second, identify the x and y coordinates of the point on the terminal side. These coordinates are the foundation for calculating the distance and the trigonometric ratios. Third, calculate the distance r from the origin to the point using the Pythagorean theorem. This distance is always positive and serves as the denominator in the sine and cosine functions. Fourth, apply the definition of sine as the ratio of the y-coordinate to the distance r. Ensure you pay attention to the sign of the y-coordinate, as it determines the sign of the sine function. Finally, compare your calculated value with the given options and select the correct answer. By following these steps consistently, you can confidently solve problems involving the terminal side of an angle and its trigonometric functions.

Terminal side, angle, standard position, sine, $\sin \theta$, trigonometry, coordinates, distance, Pythagorean theorem, Cartesian plane