Problem 63 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I noticed that for a right triangle, the Law of Cosines reduces to the Pythagorean Theorem.

Step-by-Step Solution

Answer

Yes, the statement makes sense. For a right triangle, the Law of Cosines does reduce to the Pythagorean Theorem.

Step 1 Verify the Statement

Write down the Law of Cosines formula and Pythagorean Theorem. Law of Cosines is \(c^2 = a^2 + b^2 - 2ab \cos(C)\) and Pythagorean Theorem is \(c^2 = a^2 + b^2\).

Step 2 Adjust the law of Cosines

Since we are dealing with a right triangle for this case, angle C would be 90 degrees. Plug 90 degrees into cos(C) in the Law of Cosines. The cosine of 90 degrees equals 0.

Step 3 Observe

Once 0 is plugged in, the Law of Cosines \(c^2 = a^2 + b^2 - 2ab \cos(C)\) becomes \(c^2 = a^2 + b^2 - 2ab(0)\). Since anything multiplied by 0 is 0, the equation simplifies to \(c^2 = a^2 + b^2\).

Step 4 Compare

Notice the Law of Cosines has reduced to the Pythagorean Theorem, as the statement suggested.