Find the measure of the acute angle that the given line forms with a horizontal line.
1. Y=5x+3
2. Y=1/2x+4
1. tanθ = 5
θ = 78.7°
2. tanθ = .5
θ = 26.6°
To find the measure of the acute angle that a given line forms with a horizontal line, we need to determine the slope of the line. The slope of a line represents the change in y-coordinates (vertical change) divided by the change in x-coordinates (horizontal change).
1. For the line equation, Y = 5x + 3:
The equation is already in slope-intercept form, y = mx + b, where m represents the slope of the line. In this case, the coefficient of x, 5, is the slope.
Since the slope is positive, the line is increasing as x increases, forming an acute angle with the horizontal line. To find the measure of the acute angle, we can calculate the arctangent of the slope:
Angle = arctan(5)
The result will give us the angle in radians. To convert it to degrees, we can multiply it by 180/π, approximately 57.3 degrees.
2. For the line equation, Y = (1/2)x + 4:
Again, the equation is already in slope-intercept form, y = mx + b. The coefficient of x, 1/2, represents the slope.
Similar to the previous example, the slope is positive, indicating an acute angle with the horizontal line. To find the measure of the acute angle, we calculate:
Angle = arctan(1/2)
Once again, this will give us the angle in radians. We can convert it to degrees by multiplying by 180/π, approximately 28.6 degrees.
Therefore, the measure of the acute angle formed by the line Y = 5x + 3 with a horizontal line is approximately 57.3 degrees, while the measure of the acute angle formed by the line Y = (1/2)x + 4 is approximately 28.6 degrees.