Determine the angle between the lines y = 2x + 3 and y = -1/2x + 1.

Determine the angle between the lines y = 2x + 3 and y = -1/2x + 1.

Step 1: Identify the equations of the lines. The first line is y = 2x + 3 and the second line is y = -1/2x + 1.
Step 2: Find the slope of the first line (m1). The slope is the coefficient of x, which is 2.
Step 3: Find the slope of the second line (m2). The slope is the coefficient of x, which is -1/2.
Step 4: Use the formula to find the angle θ between the two lines: θ = tan^(-1) |(m1 - m2)/(1 + m1*m2)|.
Step 5: Substitute the values of m1 and m2 into the formula: θ = tan^(-1) |(2 - (-1/2))/(1 + 2*(-1/2))|.
Step 6: Simplify the expression: θ = tan^(-1) |(2 + 1/2)/(1 - 1)|. Note that 1 - 1 = 0, so we need to be careful here.
Step 7: Calculate the numerator: 2 + 1/2 = 5/2.
Step 8: Since the denominator is 0, we need to use the slopes directly to find the angle using the tangent formula: θ = tan^(-1)(5/4).
Step 9: Calculate θ using a calculator or trigonometric tables to find that θ is approximately 60 degrees.

Slope of a Line – Understanding how to find the slope from the equation of a line in slope-intercept form (y = mx + b).
Angle Between Lines – Using the formula for the angle between two lines based on their slopes.
Trigonometric Functions – Applying the arctangent function to find the angle from the tangent value.