f a point on the Cartesian plane lies at (4, 2), what is the angle made between the line containing the point and the origin, and the negative *y*-axis?

Question 13 options:

a)

0.523 radians

b)

1.249 radians

c)

1.047 radians

d)

0.463 radians

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.

If a point on the Cartesian plane lies at (4, 2), what is the angle made between the line containing the point and the origin, and the negative y-axis? Question 13 options: a) 0.523 radians b) 1.249 radians c) 1.047 radians d) 0.463 radians

Discover how to calculate the angle made by the line passing through the origin and point (4, 2) with the negative y-axis. This solution uses key trigonometric concepts and provides step-by-step guidance. Suitable for Grades 9-11.

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