find an equation of the line passing throug (2,6) and is inclined at an angle 60 degrees counterclockwise from the horizontal
You should have learned that the slope of a line is equal to the tangent of the angle that line makes with the x-axis
so slope = tan 60° = √3/1 = √3
so the equation is:
y - 6 = √3(x - 2)
y = √3 x + 6 - 2√3
To find an equation of a line passing through a given point and inclined at a specific angle, you can follow these steps:
Step 1: Determine the slope of the line.
- The given angle is 60 degrees counterclockwise from the horizontal. This means the line makes an angle of 30 degrees with the positive x-axis.
- The slope of a line inclined at an angle θ with the positive x-axis is given by the tangent of that angle: m = tan(θ).
- In this case, the angle is 30 degrees, so the slope of the line is m = tan(30°).
Step 2: Use the point-slope form of a line to find the equation.
- The point-slope form of a line is y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope.
- Substituting the coordinates (x₁, y₁) = (2, 6) and the slope m = tan(30°), we can write the equation:
y - 6 = tan(30°)(x - 2).
This is the equation of the line passing through the point (2, 6) and inclined at an angle of 60 degrees counterclockwise from the horizontal.