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.