Find the exact value of \( \boldsymbol{\operatorname { s i n }} \mathbf{7 5}^{\circ} \)

To find the exact value of \( \sin 75^\circ \), we can use the angle addition formula. Recognizing that \( 75^\circ \) can be expressed as \( 45^\circ + 30^\circ \), we can apply the formula: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \] Setting \( a = 45^\circ \) and \( b = 30^\circ \): \[ \sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \] Now we can substitute the known values: - \( \sin 45^\circ = \frac{\sqrt{2}}{2} \) - \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) - \( \cos 45^\circ = \frac{\sqrt{2}}{2} \) - \( \sin 30^\circ = \frac{1}{2} \) Plugging these values into our expression: \[ \sin 75^\circ = \left( \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} \right) + \left( \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \right) \] Calculating the products: \[ \sin 75^\circ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \] Thus, the exact value of \( \sin 75^\circ \) is \[ \boxed{\frac{\sqrt{6} + \sqrt{2}}{4}} \]