Problem 19 - Precalculus Essentials

Question

Find the exact value of each expression. $$\cos 75^{\circ}$$

Step-by-Step Solution

Answer

The exact value of $\cos 75^{\circ}$ is $\sqrt{6}/4 - \sqrt{2}/4$

Step 1: Writing as sum of angles

Express 75 degrees as a sum of two angles whose cosine values are known. In this case, 75 degrees can be expressed as: \(75^{\circ} = 45^{\circ} + 30^{\circ}\)

Step 2: Applying identity

Apply the cosine sum of angles identity. This identity is: \(\cos(a + b) = \cos a \cos b - \sin a \sin b\) Substituting \(a = 45^{\circ}\) and \(b = 30^{\circ}\) into this identity, we get: \(\cos 75^{\circ} = \cos (45^{\circ} + 30^{\circ}) = \cos 45^{\circ} \cos 30^{\circ} - \sin 45^{\circ} \sin 30^{\circ}\)

Step 3: Substituting known values

Substitute the known sine and cosine values. The exact values are \(\cos 45^{\circ} = \sin 45^{\circ} = \sqrt{2}/2\) and \(\cos 30^{\circ} = \sqrt{3}/2\), \(\sin 30^{\circ} =1/2\). We then get: \(\cos 75^{\circ} = \sqrt{2}/2 * \sqrt{3}/2 - \sqrt{2}/2 * 1/2\)

Step 4: Simplifying

Simplify the above expression to get the final answer. This results in: \(\cos 75^{\circ} = \sqrt{6}/4 - \sqrt{2}/4\)