Problem 16 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The exact value of \(\sin 75^{\circ}\) is \(\frac{\sqrt{2} + \sqrt{6}}{4}\).

Step 1: Break Down the Angle

Break 75 degrees down into the sum of two angles. For example, it can be written as \(45^{\circ} + 30^{\circ}\).

Step 2: Substitute the Values

Substitute the values into the sum of angles identity: \(\sin(45^{\circ} + 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ}\).

Step 3: Find the Sine and Cosine Values

We know that \(\sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2}\), \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\), and \(\sin 30^{\circ} = \frac{1}{2}\). Substitute these values into the equation.

Step 4: Simplify the Expression

Simplify the expression and the final result will be \(\frac{\sqrt{2} + \sqrt{6}}{4}\).