You find the measure of a slope angle by using the arctangent (inverse tangent) function on the slope value. The slope angle is the angle the line makes with the horizontal axis.
Here's a more detailed explanation:
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Understanding Slope: The slope (often denoted as m) represents the steepness of a line. It's calculated as the change in vertical distance (rise) divided by the change in horizontal distance (run) between two points on the line:
slope (m) = rise / run. -
Relating Slope to Angle: The slope is equal to the tangent of the angle (θ) that the line makes with the x-axis. That is,
m = tan(θ). -
Finding the Angle: To find the angle θ, you need to take the inverse tangent (arctangent) of the slope:
θ = arctan(m)orθ = tan⁻¹(m). This will give you the angle in radians or degrees, depending on your calculator or software settings.
Steps to Calculate the Slope Angle:
- Determine the Slope (m): Calculate the slope of the line using two points on the line or obtain it from the line's equation (y = mx + b, where m is the slope).
- Apply the Arctangent Function: Use a calculator or software to find the arctangent (tan⁻¹) of the slope value. Ensure your calculator is set to the desired angle unit (degrees or radians).
- Interpret the Result: The result of the arctangent function is the angle in degrees or radians. This is the angle the line makes with the x-axis.
Example:
If the slope (m) of a line is 1, then the slope angle (θ) is:
θ = arctan(1) = 45 degrees (or π/4 radians)
Important Considerations:
- Calculator Settings: Be mindful of whether your calculator or software is set to degrees or radians mode, as this will affect the result.
- Quadrants: The arctangent function has a range that typically spans from -90 degrees to +90 degrees (or -π/2 to +π/2 radians). If you need to determine the angle for a line in a different quadrant, you may need to add 180 degrees (or π radians) to the result.
