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To find the angle of inclination of a line given its gradient (or slope), we use the formula:

\[
\theta = \tan^{-1}(m)
\]

Where:
- \(\theta\) is the angle of inclination.
- \(m\) is the gradient (or slope) of the line.

For each given value of \(m\), we will calculate the angle using the inverse tangent function, \(\tan^{-1}\), and round the result to the nearest degree.

### Let's calculate for each part:

1. **Part a (m = 3):**
   \[
   \theta_a = \tan^{-1}(3) \approx 71.57^\circ \quad \Rightarrow \theta_a \approx 72^\circ
   \]

2. **Part b (m = \frac{1}{2}):**
   \[
   \theta_b = \tan^{-1}\left(\frac{1}{2}\right) \approx 26.57^\circ \quad \Rightarrow \theta_b \approx 27^\circ
   \]

3. **Part c (m = 1):**
   \[
   \theta_c = \tan^{-1}(1) = 45^\circ
   \]

4. **Part d (m = 2.5):**
   \[
   \theta_d = \tan^{-1}(2.5) \approx 68.20^\circ \quad \Rightarrow \theta_d \approx 68^\circ
   \]

5. **Part e (m = -2):**
   \[
   \theta_e = \tan^{-1}(-2) \approx -63.43^\circ \quad \Rightarrow \theta_e \approx -63^\circ
   \]
   (The negative sign indicates the line is inclined downward.)

6. **Part f (m = \frac{3}{4}):**
   \[
   \theta_f = \tan^{-1}\left(\frac{3}{4}\right) \approx 36.87^\circ \quad \Rightarrow \theta_f \approx 37^\circ
   \]

7. **Part g (m = -\frac{1}{10}):**
   \[
   \theta_g = \tan^{-1}\left(-\frac{1}{10}\right) \approx -5.71^\circ \quad \Rightarrow \theta_g \approx -6^\circ
   \]

8. **Part h (m = -\frac{2}{3}):**
   \[
   \theta_h = \tan^{-1}\left(-\frac{2}{3}\right) \approx -33.69^\circ \quad \Rightarrow \theta_h \approx -34^\circ
   \]

### Summary of angles (rounded to the nearest degree):
- **a:** \(72^\circ\)
- **b:** \(27^\circ\)
- **c:** \(45^\circ\)
- **d:** \(68^\circ\)
- **e:** \(-63^\circ\)
- **f:** \(37^\circ\)
- **g:** \(-6^\circ\)
- **h:** \(-34^\circ\)

Would you like more details on any part or have any specific questions?

---

Here are 5 related questions to practice:
1. How would the angle change if the gradient was doubled for each case?
2. What happens to the angle when the slope is zero?
3. How can you graphically interpret a negative gradient's inclination?
4. How would you calculate the angle if given an inclination instead of a gradient?
5. What is the relation between the slope of a line and its perpendicular slope?

**Tip:** For gradients less than 1 (fractional), the angle of inclination is always smaller than 45 degrees.

Find, correct to the nearest degree, the angle of inclination of a line with gradient: a) 3, b) 1/2, c) 1, d) 2.5, e) -2, f) 3/4, g) -1/10, h) -2/3.

Learn how to calculate the angle of inclination of a line given its gradient using the inverse tangent function. This problem involves trigonometry and calculating the slope for gradients 3, 1/2, 1, 2.5, -2, 3/4, -1/10, and -2/3. Suitable for students in Grades 9-12.

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