Problem 20 - Precalculus Essentials

Question

Find the exact value of each expression. $$\cos 105^{\circ}$$

Step-by-Step Solution

Answer

\(\cos(105^{\circ}) = (\sqrt{2} - \sqrt{6})/4\).

Step 1: Express 105 degrees in terms of standard angles

105 degrees can be expressed as the sum of 60 degrees and 45 degrees, both of which are standard angles in the unit circle. So, we have \(105^{\circ}\) = \(60^{\circ} + 45^{\circ}\)

Step 2: Apply the formula for cosine of a sum

Using the trigonometric identity for the cosine of a sum of two angles, we get: \(\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b)\). Substituting \(a = 60^{\circ}\) and \(b = 45^{\circ}\), we get: \(\cos(105^{\circ}) = \cos(60^{\circ})\cos(45^{\circ}) - \sin(60^{\circ})\sin(45^{\circ})\).

Step 3: Calculate the final result

We know that \(\cos(60^{\circ}) = 1/2\), \(\cos(45^{\circ}) = \sqrt{2}/2\), \(\sin(60^{\circ}) = \sqrt{3}/2\), and \(\sin(45^{\circ}) = \sqrt{2}/2\). So, substituting these values into the equation, we find: \(\cos(105^{\circ}) = (1/2) * (\sqrt{2}/2) - (\sqrt{3}/2) * (\sqrt{2}/2) = \sqrt{2}/4 - \sqrt{6}/4 = (\sqrt{2} - \sqrt{6})/4\).