Final answer:
The Law of Cosines formula is c² = a² + b² - 2ab cos(C). It extends the Pythagorean theorem to all triangles by including an angle and is critical in solving for unknown sides and angles of a triangle.
Step-by-step explanation:
The student asked for the missing parts of the Law of Cosines equation which properly reads: c² = a² + b² - 2ab cos(C). It's an extension of the Pythagorean theorem applied to any triangle, not just a right triangle. If we have a triangle ABC with sides A, b, and c opposite to respective angles A, B, and C, this law relates the lengths of the sides to one of the angles. It's particularly useful for finding a side when we know the other two sides and the included angle, or finding an angle when we know all the sides.
To understand the Law of Cosines, one must first grasp the Pythagorean theorem, which applies only to right triangles and states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b) as presented in the equation: a² + b² = c². Since the Law of Cosines is a generalization when the angle C is a right angle (90 degrees), the cosine of C is zero, and the law simplifies to the Pythagorean theorem.
This law is vital for solving triangles in trigonometry, and it's a fundamental concept to grasp for students progressing in mathematics. When applied, one can calculate unknown distances or angles, making it essential in fields such as engineering, physics, and even geography.