The given triangle is an oblique triangle, where we have:

• Side $a = 192$
• Side $b = 122$
• Angle $C = 22.2^\circ$

To solve the triangle, we need to find the missing sides $c$ and the angles $A$ and $B$.

Step 1: Use the Law of Cosines

Since we have two sides and an included angle (SAS case), we can use the Law of Cosines to find the third side $c$:

$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$

Substituting the given values:

$c^2 = 192^2 + 122^2 - 2(192)(122) \cdot \cos(22.2^\circ)$

$c^2 = 36864 + 14884 - 2(192)(122) \cdot 0.92718$

$c^2 = 36864 + 14884 - 43262.5$

$c^2 = 51748 - 43262.5 = 8485.5$

$c = \sqrt{8485.5} \approx 92.1$

So, the third side is $c \approx 92$.

Step 2: Use the Law of Sines to Find Angle $A$

Now that we know all three sides, we can use the Law of Sines to find the other angles. The Law of Sines states:

$\frac{\sin A}{a} = \frac{\sin C}{c}$

Rearranging to solve for $A$:

$\sin A = \frac{a \cdot \sin C}{c}$

Substitute the known values:

$\sin A = \frac{192 \cdot \sin(22.2^\circ)}{92}$

$\sin A = \frac{192 \cdot 0.377}{92} = \frac{72.38}{92} = 0.7867$

Now, take the inverse sine:

$A = \arcsin(0.7867) \approx 51.7^\circ$

So, angle $A \approx 52^\circ$.

Step 3: Find Angle $B$

The sum of the angles in any triangle is $180^\circ$. So, to find angle $B$, we subtract the known angles $A$ and $C$ from $180^\circ$:

$B = 180^\circ - A - C = 180^\circ - 52^\circ - 22.2^\circ = 105.8^\circ$

So, angle $B \approx 106^\circ$.

Summary of Results:

• Side $c \approx 92$
• Angle $A \approx 52^\circ$
• Angle $B \approx 106^\circ$

Let me know if you would like further details or clarifications!

Here are some related questions you might explore:

1. What happens if one of the sides is larger than the sum of the other two?
2. How can we verify the solution using only the Law of Cosines?
3. How would the process change if all angles were given, but no sides?
4. How does rounding intermediate results affect the final answers?
5. How can the Law of Sines help solve ambiguous cases (SSA)?

Tip: Always double-check the sum of angles in a triangle to ensure they add up to $180^\circ$.