Welcome to our exploration of the Law of Cosines. This important theorem is a generalization of the Pythagorean Theorem that works for any triangle, not just right triangles. It helps us solve triangles in two key scenarios: when we know two sides and the included angle, or when we know all three sides. The Law of Cosines states that c squared equals a squared plus b squared minus two times a times b times the cosine of angle C. This formula relates the sides and angles of any triangle.
The Law of Cosines can be written in three equivalent forms, depending on which side and angle we're focusing on. The first form states that c squared equals a squared plus b squared minus two a b cosine C. Similarly, we can write that a squared equals b squared plus c squared minus two b c cosine A. And the third form is b squared equals a squared plus c squared minus two a c cosine B. Notice the pattern: each form solves for one side squared in terms of the other two sides and the angle opposite to the side we're solving for. As a special case, when one of the angles is 90 degrees, the cosine term becomes zero, and we get the familiar Pythagorean Theorem: c squared equals a squared plus b squared.
Let's prove the Law of Cosines. We start with a triangle ABC and draw an altitude h from vertex C to side AB. Let's call the foot of this altitude point D. This creates two right triangles: ADC and BDC. If we let AD equal x, then DB equals c minus x. Now we can apply the Pythagorean Theorem to both right triangles. In triangle ADC, we get b squared equals h squared plus x squared. In triangle BDC, we get a squared equals h squared plus the quantity c minus x squared. From the first equation, we know that h squared equals b squared minus x squared. Substituting this into the second equation and expanding, we get a squared equals b squared minus x squared plus c squared minus 2cx plus x squared. The x squared terms cancel out, giving us a squared equals b squared plus c squared minus 2cx. Now, in triangle ADC, by the definition of cosine, we know that cosine of angle A equals x divided by b, which means x equals b times cosine A. Substituting this into our equation, we get a squared equals b squared plus c squared minus 2c times b cosine A, which is the Law of Cosines.
The Law of Cosines has two main applications in solving triangles. First, in the Side-Angle-Side case, when we know two sides and the included angle, we can directly find the third side using the formula. For example, if we know sides b and c, and the angle A between them, we can calculate side a using the formula: a squared equals b squared plus c squared minus two b c cosine A. Second, in the Side-Side-Side case, when we know all three sides of a triangle, we can find any angle using the rearranged formula. For instance, to find angle A, we use: cosine A equals b squared plus c squared minus a squared, all divided by two b c. Then we apply the inverse cosine function to find the angle. These applications make the Law of Cosines an essential tool for solving triangles in geometry, trigonometry, and real-world problems.
To summarize what we've learned about the Law of Cosines: First, it's a generalization of the Pythagorean Theorem that works for any triangle, not just right triangles. The formula states that c squared equals a squared plus b squared minus two a b times the cosine of angle C. We use this law in two main scenarios: when we have two sides and the included angle (SAS) to find the third side, and when we have all three sides (SSS) to find any angle. The Law of Cosines is essential for solving non-right triangles in geometry, trigonometry, physics, engineering, and many real-world applications where the Pythagorean Theorem alone is insufficient.