Problem 15 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The exact value of \(\sin 105^{\circ}\) is \(\frac{\sqrt{6} + \sqrt{2}}{4}\).

Step 1: Express the angle as sum of two angles

Write \(105^{\circ}\) as a sum of two angles that occur in the unit circle, i.e. \(60^{\circ} + 45^{\circ}\).

Step 2: Apply the trigonometric formula

Apply the trigonometric formula for sine of sum of two angles: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\). For \(60^{\circ} + 45^{\circ}\), it becomes \(\sin(60^{\circ} + 45^{\circ}) = \sin 60^{\circ} \cos 45^{\circ} + \cos 60^{\circ} \sin 45^{\circ}\).

Step 3: Look up trigonometric values and substitute

Replace the values of \(\sin 60^{\circ}\), \(\sin 45^{\circ}\), \(\cos 60^{\circ}\) and \(\cos 45^{\circ}\) from the unit circle with their corresponding values: \(\frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2}\), respectively. So we get \(\sin 105^{\circ} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}+ \frac{1}{2} \cdot \frac{\sqrt{2}}{2}\).

Step 4: Simplify the expression

Multiplication of the terms yields: \(\sin 105^{\circ} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}\). Further simplification can be achieved by factoring out \(\frac{1}{4}\) to get: \(\sin 105^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4}\).