To solve this problem, we calculate the angle $\theta$ in radians for each point using the formula:

$\theta = \arctan\left(\frac{y}{x}\right)$

Key Considerations:

1. The quadrant determines the angle's sign and the range of $\theta$.
2. Adjust $\theta$ as needed to ensure it lies within $[0, 2\pi)$ for standard position.

Calculations:

Let's compute each one.

a) $(-3, 14)$

• $x = -3, y = 14$ $\theta = \arctan\left(\frac{14}{-3}\right) = \arctan(-4.67)$ This point is in the second quadrant, so: $\theta = \pi + \arctan(-4.67) \approx 1.93 \, \text{radians}.$

b) $(6, 7)$

• $x = 6, y = 7$ $\theta = \arctan\left(\frac{7}{6}\right) \approx \arctan(1.167) \approx 0.86 \, \text{radians}.$

c) $(1, 9)$

• $x = 1, y = 9$ $\theta = \arctan\left(\frac{9}{1}\right) = \arctan(9) \approx 1.46 \, \text{radians}.$

d) $(-5, -18)$

• $x = -5, y = -18$ $\theta = \arctan\left(\frac{-18}{-5}\right) = \arctan(3.6)$ This point is in the third quadrant, so: $\theta = \pi + \arctan(3.6) \approx 4.25 \, \text{radians}.$

e) $(2, 3)$

• $x = 2, y = 3$ $\theta = \arctan\left(\frac{3}{2}\right) = \arctan(1.5) \approx 0.98 \, \text{radians}.$

f) $(4, -20)$

• $x = 4, y = -20$ $\theta = \arctan\left(\frac{-20}{4}\right) = \arctan(-5)$ This point is in the fourth quadrant, so: $\theta = 2\pi + \arctan(-5) \approx 4.92 \, \text{radians}.$

Final Results (to nearest hundredth):

a) $1.93 \, \text{radians}$
b) $0.86 \, \text{radians}$
c) $1.46 \, \text{radians}$
d) $4.25 \, \text{radians}$
e) $0.98 \, \text{radians}$
f) $4.92 \, \text{radians}$

Would you like step-by-step verification or further clarification?

Related Questions:

1. How is the arctangent function modified in different quadrants?
2. Why is the angle shifted by $\pi$ in the second and third quadrants?
3. How can this process be applied to find angles for 3D coordinates?
4. What are the properties of the tangent function used in these calculations?
5. How can calculators be used efficiently to find angles in radians?

Tip: Always verify which quadrant a point lies in before applying trigonometric functions.