To solve for $x^\circ$ in the triangle:

1. Step 1: Recall the sum of angles in a triangle The sum of the angles in any triangle is $180^\circ$: $x + 59^\circ + \text{other angle} = 180^\circ$

2. Step 2: Find the other angle Using the two sides of the triangle, $26$ and $30$, and the included angle $59^\circ$, we use the law of cosines to find the third side, followed by the third angle.

   Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos(C)$ Substituting $a = 26$, $b = 30$, and $C = 59^\circ$: $c^2 = 26^2 + 30^2 - 2(26)(30)\cos(59^\circ)$

   Calculate $c^2$, then use the law of sines to find the other angle.

   Let math

Given values

a = 26 # side a b = 30 # side b C = math.radians(59) # included angle in radians

Applying the Law of Cosines to find side c

c_squared = a2 + b2 - 2 * a * b * math.cos(C) c = math.sqrt(c_squared) # side opposite the given angle

Using Law of Sines to find the other angle

sin(A)/a = sin(C)/c

sin_A = (a * math.sin(C)) / c A = math.degrees(math.asin(sin_A)) # angle opposite side a

Finding the remaining angle x

x = 180 - 59From the calculations:

1. The third side of the triangle (opposite $59^\circ$) is approximately $27.79$.
2. The angle opposite side $26$ is approximately $53.30^\circ$.
3. The angle $x$ is therefore: $x \approx 180^\circ - 59^\circ - 53.30^\circ = 67.70^\circ$

Thus, $x \approx 67.70^\circ$.

Would you like further details on any step, or do you have additional questions? Here are five questions related to this:

1. How is the Law of Cosines derived and applied?
2. Can we use the Law of Sines to directly solve for $x$?
3. What happens if a triangle has angles summing to more than (180^\