Problem 105 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a half-angle formula to find the exact value of \(\cos 100^{\circ}\)

Step-by-Step Solution

Answer

The statement doesn't make sense because half-angle identities are typically used to find exact values for half of known angles, and 100 degrees is not half of any common angles.

Step 1: Understanding the Statement

The statement says that a half-angle formula was used to find the exact value of \(\cos 100^{\circ}\). It needs to be determined if this approach makes sense or not.

Step 2: Recalling Half-Angle Formulas

Recall the half-angle identity for cosine, \(\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}.\) This identity allows for the exact calculation of the half angles of known angles.

Step 3: Analyzing the Statement

\(\cos 100^{\circ}\) is not a half-angle of any commonly known angles (30, 45, 60, 90,180). Hence, we may say that it is not reasonable to use the half-angle formula to find \(\cos 100^{\circ}\) exactly, since we will not have a simple form for \(2 \times 100 = 200^{\circ}\), outside the standard unit circle angles. We could technically use the half-angle formula, but it wouldn't simplify the process.