Determine the Equation of a Straight Line Given Its Slope and a Point

In this article, we will explore how to derive the equation of a straight line that passes through a given point and makes a specific angle with the positive x-axis. We will start by understanding the relationship between the angle's sine and cosine, followed by illustrating a step-by-step method to find the line's equation. This process is essential for grasping the principles of linear functions and their applications in various fields such as physics, engineering, and data science.

Introduction to the Problem

We are tasked with finding the equation of a straight line that passes through the point (1, 2) and makes an angle with the positive x-axis such that the sine of this angle is 3/5.

Step 1: Determine the Angle

Given that the sine of the angle θ is 3/5, we can use a trigonometric identity to determine the cosine of the angle. The relationship between sine and cosine in a right-angled triangle is given by: [ sin^2theta cos^2theta 1. ] Plugging in the value of sinθ, we get: [ left(frac{3}{5}right)^2 cos^2theta 1 rightarrow frac{9}{25} cos^2theta 1. ] From this, we can solve for cosθ:

[ cos^2theta 1 - frac{9}{25} frac{16}{25}, ]

and therefore:

[ costheta pm frac{4}{5}. ]

Step 2: Determine the Slope (m)

The slope (m) of the line can be calculated using the tangent of the angle, as the slope is the ratio of the opposite side to the adjacent side in a right-angled triangle. Thus:

[ m tantheta frac{sintheta}{costheta}. ]

For cosθ 4/5:

[ m frac{frac{3}{5}}{frac{4}{5}} frac{3}{4}. ]

For cosθ -4/5:

[ m frac{frac{3}{5}}{-frac{4}{5}} -frac{3}{4}. ]

Step 3: Equation of the Line

Using the point-slope form of the line equation, ( y - y_1 m(x - x_1) ), where the point (x?, y?) is (1, 2) and the slope (m) is as determined above, we can derive the equations of the lines:

For m 3/4: [ y - 2 frac{3}{4}(x - 1) rightarrow 4y - 8 3x - 3 rightarrow 4y 3x 5. ] For m -3/4: [ y - 2 -frac{3}{4}(x - 1) rightarrow 4y - 8 -3x 3 rightarrow 4y -3x 11. ]

Thus, the equations of the lines are:

For (m frac{3}{4}): [ y frac{3}{4}x frac{5}{4}. ] For (m -frac{3}{4}): [ y -frac{3}{4}x frac{11}{4}. ]

These are the two possible lines that pass through point (1, 2) and satisfy the given angle condition.

Conclusion

This analysis demonstrates the process of finding the equation of a straight line given its slope and a point, and how to use trigonometric functions to determine the line's slope based on a given angle. The equations derived provide a clear expression of the lines in question, which can be utilized in various mathematical and practical applications.