Here are solutions to some of the math problems from the document:

1. Right Triangle Problem (#19)

You need to solve a right triangle with given side lengths or angles. Without specific details from the triangle, the general approach would be:

• Use Pythagoras' theorem for missing sides: $c^2 = a^2 + b^2$
• Use trigonometric ratios to find angles:
  • $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
  • $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
  • $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

2. Solve $\sin A = 0.3746$ (#20)

To find angles $A$ between $0^\circ$ and $360^\circ$:

1. Find the reference angle: $A_1 = \arcsin(0.3746) \approx 22^\circ$
2. The possible solutions are: $A_1 = 22^\circ \quad \text{and} \quad A_2 = 180^\circ - 22^\circ = 158^\circ$ Thus, $A = 22^\circ, 158^\circ$.

3. Use Law of Sines to Find Side $AB$ (#22)

The Law of Sines is given by: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ Use the known angles and one side to find the unknown side.

4. Use Law of Cosines to Find Side $AB$ (#23)

The Law of Cosines is: $c^2 = a^2 + b^2 - 2ab \cos C$ Plug in the known values to solve for $c$.

5. Volume of a Cone Inscribed in a Sphere (#39a)

The volume of a cone inscribed in a sphere can be shown using geometry and calculus: $V = \frac{\pi}{3} h (2R^2 - h^2)$

Would you like details on any of the problems above, or a solution to a specific question?

Here are five additional questions you might explore:

1. How can the Law of Sines be derived?
2. What are the uses of trigonometric identities in solving triangles?
3. How does one compute the volume of irregular geometric shapes?
4. What are some applications of the Law of Cosines in real-world problems?
5. How do you solve for trigonometric angles using inverse functions?

Tip: When solving for angles with trigonometric functions, always check if the angle is in the correct quadrant!