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Calculating the angles of a triangle is a fundamental aspect of trigonometry and geometry. When we know all three side lengths of a triangle, as given in our exercise with sides of 9 feet, 5 feet, and 7 feet, we can use the Law of Cosines to find the angles. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. This law is given by the formula \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \), where \( a \) and \( b \) are the lengths of the sides adjacent to angle \( C \), and \( c \) is the length of the opposite side.

By rearranging the formula, we can solve for \( \cos(C) \) and then determine angle \( C \) using its inverse cosine. This process can be repeated for the other two angles, ensuring that we work methodically to accurately determine each angle. Remember, in a triangle, the sum of the interior angles always equals 180 degrees. Thus, after finding two angles, we can simply subtract their sum from 180 to find the third angle. This final step serves as a check and confirms that the calculations were performed correctly.

The inverse cosine function, denoted as \( \cos^{-1} \), is used to find an angle whose cosine is a given number. For example, if we have a value of \( \cos(\theta) = 0.5 \), then \( \theta = \cos^{-1}(0.5) \). It's important to note that the output of \( \cos^{-1} \) is in the range from 0 to 180 degrees or 0 to \( \pi \) radians when dealing with real numbers, which matches the range of possible angles in a triangle.

In our exercise, we applied the inverse cosine function to find angles \( A \) and \( B \) by evaluating \( \cos^{-1}\left(-\frac{1}{2}\right) \) and \( \cos^{-1}\left(\frac{7}{6}\right) \) respectively. This step is essential after calculating the cosine of the angle using the Law of Cosines—without the inverse cosine function, we would only have the value of cosine, not the actual angle measurement.

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The Law of Cosines is an important trigonometric identity that extends the Pythagorean theorem to non-right triangles. Trigonometry also provides other functions, such as sine and tangent, which along with cosine, help in solving for unknowns in triangles.

Understanding how to manipulate these functions is crucial for solving many geometric problems, like those involving triangle angle calculations. The trigonometric functions are periodic and analytical, which means that they repeat values over intervals and can be expressed as an infinite sum of terms, respectively. Mastery of trigonometry is not just about memorizing formulas, but also about understanding the underlying principles and relationships that govern these formulas and their applications in various contexts.