Find the angle between the lines y = 2x + 1 and y = -0.5x + 3.
Correct Answer: 90 degrees
- Step 1: Identify the equations of the lines. The first line is y = 2x + 1 and the second line is y = -0.5x + 3.
- 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 -0.5.
- Step 4: Use the formula to find the angle θ between the two lines: tan(θ) = |(m1 - m2) / (1 + m1*m2)|.
- Step 5: Substitute the values of m1 and m2 into the formula: tan(θ) = |(2 - (-0.5)) / (1 + 2 * -0.5)|.
- Step 6: Simplify the expression: tan(θ) = |(2 + 0.5) / (1 - 1)|.
- Step 7: Notice that the denominator (1 - 1) equals 0, which means tan(θ) is undefined.
- Step 8: When tan(θ) is undefined, it indicates that the angle θ is 90 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 Two Lines – Using the formula for the angle between two lines based on their slopes.
- Undefined Values in Trigonometry – Recognizing when a tangent function is undefined and what that implies about the angle.
