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Solving triangles is the process of determining the unknown lengths and angles of a triangle when given some initial information. It's a bit like being a detective, gathering clues and piecing them together to form a complete picture.

In the case where we have a mix of angles and sides provided, like in the problem statement with sides a, c and angle A, we use specific formulas to find the missing pieces. This is often done through the application of the Law of Cosines or Law of Sines, depending on what information we are given and what we need to find out.

Once we find one length or angle, it can unlock the next piece of the puzzle, and we continue this process until we have found all the missing information. By the end, all three sides and angles of the triangle are known, completing the triangle's 'portrait'.

The Law of Sines is a powerful tool in the world of trigonometry, especially when it comes to triangles. It states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a triangle. The formula looks like this:
\[\begin{equation}\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\end{equation}\]
When we have one side length and its opposite angle, we can use this formula to find out other angle-side opposite pairs.

For example, in the textbook problem, once we've calculated the length of side b using the Law of Cosines, we can find the angle B by rearranging the Law of Sines formula. It's like unlocking a new level in a game where the prize is the value of angle B!

The realm of trigonometric functions is vast and essential for understanding triangles. These mathematical heroes — sine (sin), cosine (cos), and tangent (tan) — help us relate angles to side lengths. They are ratios from a right-angled triangle but extend their usefulness to any triangle and even circles!

Sine relates to the ratio of the opposite side to the hypotenuse, cosine to the adjacent side over the hypotenuse, and tangent to the opposite over the adjacent. These are just the starters! There are their inverse functions too, allowing us to work backward from the ratio to find the angle.

In the context of our triangle problem, the Law of Cosines uses the cosine function to find a missing side, and the Law of Sines uses the sine function to find a missing angle. They are the bridge connecting the known to the unknown in our geometric quest.

Every triangle, no matter its shape or size, has one thing in common: the interior angles always add up to 180 degrees. This reliable rule is like a trusty compass guiding us through the journey of solving triangles.

Imagine having a triangle consisting of angles A, B, and C. It is guaranteed that:\[\begin{equation}A + B + C = 180^{\textdegree}\end{equation}\]
Knowing just one or two angles can help us find the remaining angle. For example, if angle A and angle B are known, angle C is simply the rest of what's needed to reach 180 degrees. This concept finalizes our mission of solving the triangle in the given exercise, allowing us to calculate angle C after finding angle B using the Law of Sines. It's the final piece of the puzzle that brings the whole picture into view.