Find C-m Value Line Passes Point (3,4) At Angle Sin⁻¹(3/5)
Introduction
In the realm of coordinate geometry, the interplay between lines, points, and angles forms a fascinating area of study. This article delves into a specific problem where a line, defined by the equation y = mx + c, passes through a given point and makes a particular angle with the positive x-axis. Our objective is to determine the value of c - m, where m represents the slope of the line and c represents the y-intercept. Understanding the relationship between the slope, angle, and points on a line is crucial for solving such problems. We will explore how the angle of inclination, expressed as an inverse sine function, relates to the slope of the line and how the given point helps us establish a concrete relationship between m and c. This exploration will not only provide a solution to the problem but also enhance our understanding of the fundamental principles governing linear equations and their geometric interpretations.
Problem Statement
If the line y = mx + c passes through the point (3, 4) and makes an angle of sin⁻¹(3/5) with the positive direction of the x-axis, then what is the value of c - m?
This problem elegantly combines the concepts of linear equations, coordinate geometry, and trigonometry. The equation y = mx + c represents a straight line where m is the slope and c is the y-intercept. The point (3, 4) provides us with a specific location on this line, and the angle sin⁻¹(3/5) gives us information about the line's inclination with respect to the x-axis. The challenge lies in connecting these pieces of information to find the values of m and c and ultimately calculate c - m. To solve this, we need to understand how the angle of inclination relates to the slope of the line and how the coordinates of a point on the line satisfy its equation. The inverse sine function adds a trigonometric element to the problem, requiring us to recall the relationship between sine, cosine, and tangent, which will be crucial in determining the slope.
Solution
1. Understanding the Slope and Angle Relationship
The slope (m) of a line is directly related to the angle (θ) it makes with the positive x-axis. The relationship is given by:
m = tan(θ)
In this case, the angle θ is given as sin⁻¹(3/5). To find tan(θ), we first need to determine cos(θ). We know that:
sin²(θ) + cos²(θ) = 1
Given sin(θ) = 3/5, we can substitute this into the equation:
(3/5)² + cos²(θ) = 1
9/25 + cos²(θ) = 1
cos²(θ) = 1 - 9/25
cos²(θ) = 16/25
cos(θ) = ±4/5
Since the angle is with respect to the positive x-axis, we consider the positive value, so cos(θ) = 4/5. Now we can find tan(θ) using the relationship:
tan(θ) = sin(θ) / cos(θ)
tan(θ) = (3/5) / (4/5)
tan(θ) = 3/4
Therefore, the slope m of the line is 3/4.
2. Using the Point (3, 4) to Find c
Now that we have the slope, we can use the fact that the line passes through the point (3, 4). This means that the coordinates of this point satisfy the equation of the line y = mx + c. Substituting x = 3, y = 4, and m = 3/4 into the equation, we get:
4 = (3/4)(3) + c
4 = 9/4 + c
To solve for c, we subtract 9/4 from both sides:
c = 4 - 9/4
c = 16/4 - 9/4
c = 7/4
So, the y-intercept c is 7/4.
3. Calculating c - m
Finally, we can calculate c - m using the values we found:
c - m = 7/4 - 3/4
c - m = 4/4
c - m = 1
Therefore, the value of c - m is 1.
Conclusion
In this problem, we successfully determined the value of c - m by leveraging the relationship between the angle of inclination, the slope of a line, and the coordinates of a point lying on that line. The key steps involved finding the slope m using the given angle sin⁻¹(3/5), then using the point (3, 4) to find the y-intercept c. Finally, we calculated the difference c - m. This exercise highlights the importance of understanding fundamental concepts in coordinate geometry and trigonometry and how they can be applied to solve specific problems. The problem's solution not only provides a numerical answer but also reinforces the understanding of the interplay between geometric and algebraic representations of lines. The relationship m = tan(θ) and the substitution of point coordinates into the line equation are powerful tools in solving similar problems, offering a clear and concise method for finding unknown parameters of a line.
Keywords
Line equation, slope, y-intercept, angle of inclination, inverse sine function, coordinate geometry, trigonometry, point-slope form, linear equations, tan(θ), sin(θ), cos(θ), c - m
FAQ
Q: How does the angle of inclination relate to the slope of a line? A: The slope (m) of a line is equal to the tangent of the angle (θ) it makes with the positive x-axis, i.e., m = tan(θ).
Q: Why do we need to find cos(θ) when sin(θ) is given? A: Because the slope is calculated using tan(θ), which is sin(θ) / cos(θ). Therefore, to find tan(θ), we need both sin(θ) and cos(θ).
Q: How do we use a point on the line to find the y-intercept? A: By substituting the coordinates of the point (x, y) and the slope m into the line equation y = mx + c, we can solve for the y-intercept c.
Q: What is the significance of the value c - m? A: c - m is a specific value that combines the y-intercept and the slope of the line. Its value provides a unique characteristic of the line based on its position and orientation in the coordinate plane.
Q: Can this method be used for other similar problems? A: Yes, this method can be applied to various problems involving lines, angles, and points in coordinate geometry. The key is to understand the relationships between the given information and apply the relevant formulas and equations.
