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The cosine rule is a fundamental concept for solving triangles, particularly useful when you know all three sides but none of the angles. It allows us to find one of the angles when the other two sides and opposite side length are known. This is incredibly helpful in non-right-angled triangles.In our exercise, the cosine rule for finding angle \(A\) is given as:\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]Here, \(a\) is the side opposite angle \(A\), and \(b\) and \(c\) are the other two sides. By plugging the lengths of the sides into the formula, we can solve for \(\cos A\) and consequently use the inverse cosine function to find the angle \(A\). A critical thing to note is the relationship this rule reveals between the angles and sides, helping to deduce whether a triangle is acute, right-angled, or obtuse based on the calculated angle.

The sine rule is another important tool in triangle problem-solving, allowing us to find unknown angles or sides in a triangle when we already have some of them.The formula states:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]In the given exercise, once we've determined angle \(A\) using the cosine rule, the sine rule helps us find another angle \(B\). Knowing \(A\) and its corresponding side \(a\), we can solve for \(B\) using:\[ \sin B = \frac{b \times \sin A}{a} \]This method is particularly useful because it simplifies finding additional unknown angles when one angle and its opposite side are known.

Understanding the types of triangles is vital when solving triangle problems, as each type offers different properties and rules. Triangles can be classified based on:

• Angles: Right, acute, or obtuse.
• Sides: Equilateral, isosceles, or scalene.

In our exercise, by using the cosine rule, we initially check the triangle type based on its angles. A right triangle has one 90-degree angle, whereas an acute triangle has all angles less than 90 degrees. An obtuse triangle has one angle greater than 90 degrees. Knowing a triangle's type can determine which rules and formulas are applicable, simplifying the problem-solving process further.

The angle sum property is a simple yet powerful concept that states the sum of all interior angles in any triangle is always \(180^\circ\).In solving triangles, after calculating two angles using the cosine and sine rules, we can easily find the third angle using this property. This makes it a convenient way to complete our angle calculations.For the given problem, once we find angles \(A\) and \(B\), we compute angle \(C\) by subtracting the sum of \(A\) and \(B\) from \(180^\circ\):\[ C = 180^\circ - A - B \]This step is crucial as it ensures the triangle is correctly solved, with all angles neatly adding up to the expected total. Understanding this property not only helps compute missing angles but also verifies the accuracy of your calculations.