To solve the problem, we need to determine the angle between the line containing the point $(4, 2)$ and the origin, relative to the negative $y$-axis.

Steps:

1. Find the angle with respect to the positive $x$-axis: The point $(4, 2)$ gives us a line from the origin to this point. The angle $\theta$ with the positive $x$-axis can be calculated using the formula: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ Substituting $x = 4$ and $y = 2$: $\theta = \tan^{-1}\left(\frac{2}{4}\right) = \tan^{-1}(0.5)$ From a calculator: $\theta \approx 0.464 \, \text{radians.}$

2. Adjust for the negative $y$-axis: The angle with respect to the negative $y$-axis is obtained by subtracting the angle $\theta$ from $\frac{\pi}{2}$ (as the negative $y$-axis is perpendicular to the positive $x$-axis but in the opposite direction): \text{Angle with negative y-axis} = \frac{\pi}{2} - \theta Substituting $\theta \approx 0.464 \, \text{radians}$: \text{Angle with negative y-axis} = \frac{\pi}{2} - 0.464 \approx 1.249 \, \text{radians.}

Final Answer:

b) 1.249 radians

Let me know if you'd like a deeper explanation or calculations!

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5 Related Questions:

1. What is the distance of the point $(4, 2)$ from the origin?
2. How would the angle change if the point were $(-4, -2)$?
3. What is the slope of the line passing through the origin and the point $(4, 2)$?
4. How do you convert radians to degrees, and what is the angle in degrees?
5. What is the general formula for finding the angle between a point and any axis?

Tip:

Always sketch the situation on the Cartesian plane to visualize angles and avoid confusion about quadrant-specific adjustments.