Trigonometric equations to solve for angles between 0° and 360°.

To solve the given trigonometric equations for all solutions between \(0^\circ\) and \(360^\circ\), we will use the properties of trigonometric functions, including their periodicity, symmetry, and the unit circle. Let's solve each equation step by step.

---

a) \(\sin x = 0.5\)

The sine function is positive in the first and second quadrants. The reference angle where \(\sin x = 0.5\) is:
\[
x = \arcsin(0.5) = 30^\circ
\]
In the second quadrant, the angle is:
\[
180^\circ - 30^\circ = 150^\circ
\]
Thus, the solutions are:
\[
x = 30^\circ, 150^\circ
\]

---

b) \(\cos x = 0.5\)

The cosine function is positive in the first and fourth quadrants. The reference angle where \(\cos x = 0.5\) is:
\[
x = \arccos(0.5) = 60^\circ
\]
In the fourth quadrant, the angle is:
\[
360^\circ - 60^\circ = 300^\circ
\]
Thus, the solutions are:
\[
x = 60^\circ, 300^\circ
\]

---

c) \(\tan x = 1\)

The tangent function is positive in the first and third quadrants. The reference angle where \(\tan x = 1\) is:
\[
x = \arctan(1) = 45^\circ
\]
In the third quadrant, the angle is:
\[
180^\circ + 45^\circ = 225^\circ
\]
Thus, the solutions are:
\[
x = 45^\circ, 225^\circ
\]

---

d) \(\sin A = -0.7\)

The sine function is negative in the third and fourth quadrants. The reference angle where \(\sin A = 0.7\) is:
\[
A = \arcsin(0.7) \approx 44.4^\circ
\]
In the third quadrant, the angle is:
\[
180^\circ + 44.4^\circ = 224.4^\circ
\]
In the fourth quadrant, the angle is:
\[
360^\circ - 44.4^\circ = 315.6^\circ
\]
Thus, the solutions are:
\[
A \approx 224.4^\circ, 315.6^\circ
\]

---

e) \(\cos A = -0.2\)

The cosine function is negative in the second and third quadrants. The reference angle where \(\cos A = 0.2\) is:
\[
A = \arccos(0.2) \approx 78.5^\circ
\]
In the second quadrant, the angle is:
\[
180^\circ - 78.5^\circ = 101.5^\circ
\]
In the third quadrant, the angle is:
\[
180^\circ + 78.5^\circ = 258.5^\circ
\]
Thus, the solutions are:
\[
A \approx 101.5^\circ, 258.5^\circ
\]

---

f) \(\tan A = -3\)

The tangent function is negative in the second and fourth quadrants. The reference angle where \(\tan A = 3\) is:
\[
A = \arctan(3) \approx 71.6^\circ
\]
In the second quadrant, the angle is:
\[
180^\circ - 71.6^\circ = 108.4^\circ
\]
In the fourth quadrant, the angle is:
\[
360^\circ - 71.6^\circ = 288.4^\circ
\]
Thus, the solutions are:
\[
A \approx 108.4^\circ, 288.4^\circ
\]

---

g) \(\sin \theta = 0.75\)

The sine function is positive in the first and second quadrants. The reference angle where \(\sin \theta = 0.75\) is:
\[
\theta = \arcsin(0.75) \approx 48.6^\circ
\]
In the second quadrant, the angle is:
\[
180^\circ - 48.6^\circ = 131.4^\circ
\]
Thus, the solutions are:
\[
\theta \approx 48.6^\circ, 131.4^\circ
\]

---

h) \(\cos \theta = -0.75\)

The cosine function is negative in the second and third quadrants. The reference angle where \(\cos \theta = 0.75\) is:
\[
\theta = \arccos(0.75) \approx 41.4^\circ
\]
In the second quadrant, the angle is:
\[
180^\circ - 41.4^\circ = 138.6^\circ
\]
In the third quadrant, the angle is:
\[
180^\circ + 41.4^\circ = 221.4^\circ
\]
Thus, the solutions are:
\[
\theta \approx 138.6^\circ, 221.4^\circ
\]

---

i) \(\tan \theta = 0.05\)

The tangent function is positive in the first and third quadrants. The reference angle where \(\tan \theta = 0.05\) is:
\[
\theta = \arctan(0.05) \approx 2.86^\circ
\]
In the third quadrant, the angle is:
\[
180^\circ + 2.86^\circ = 182.86^\circ
\]
Thus, the solutions are:
\[
\theta \approx 2.86^\circ, 182.86^\circ
\]

---

j) \(\tan \theta = -\frac{4}{9}\)

The tangent function is negative in the second and fourth quadrants. The reference angle where \(\tan \theta = \frac{4}{9}\) is:
\[
\theta = \arctan\left(\frac{4}{9}\right) \approx 23.8^\circ
\]
In the second quadrant, the angle is:
\[
180^\circ - 23.8^\circ = 156.2^\circ
\]
In the fourth quadrant, the angle is:
\[
360^\circ - 23.8^\circ = 336.2^\circ
\]
Thus, the solutions are:
\[
\theta \approx 156.2^\circ, 336.2^\circ
\]

---

k) \(\sin x = 0.6\)

The sine function is positive in the first and second quadrants. The reference angle where \(\sin x = 0.6\) is:
\[
x = \arcsin(0.6) \approx 36.9^\circ
\]
In the second quadrant, the angle is:
\[
180^\circ - 36.9^\circ = 143.1^\circ
\]
Thus, the solutions are:
\[
x \approx 36.9^\circ, 143.1^\circ
\]

---

l) \(\cos x = 2\)

The cosine function has a range of \([-1, 1]\). Since \(2\) is outside this range, there are no real solutions for this equation.

---

Final Answers:

\[
\boxed{
\begin{aligned}
&\text{a) } x = 30^\circ, 150^\circ \\
&\text{b) } x = 60^\circ, 300^\circ \\
&\text{c) } x = 45^\circ, 225^\circ \\
&\text{d) } A \approx 224.4^\circ, 315.6^\circ \\
&\text{e) } A \approx 101.5^\circ, 258.5^\circ \\
&\text{f) } A \approx 108.4^\circ, 288.4^\circ \\
&\text{g) } \theta \approx 48.6^\circ, 131.4^\circ \\
&\text{h) } \theta \approx 138.6^\circ, 221.4^\circ \\
&\text{i) } \theta \approx 2.86^\circ, 182.86^\circ \\
&\text{j) } \theta \approx 156.2^\circ, 336.2^\circ \\
&\text{k) } x \approx 36.9^\circ, 143.1^\circ \\
&\text{l) No solutions}
\end{aligned}
}
\]