Calculate the following without the use of a calculato (1) \( \sin 75^{\circ} \) (3) \( \sin 50^{\circ} \cos 10^{\circ}+\cos 50^{\circ} \sin 10^{\circ} \)

To solve the problems, we will use trigonometric identities and known values. ### (1) Calculate \( \sin 75^{\circ} \) We can express \( 75^{\circ} \) as the sum of \( 45^{\circ} \) and \( 30^{\circ} \): \[ \sin 75^{\circ} = \sin(45^{\circ} + 30^{\circ}) \] Using the sine addition formula: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \] we can substitute \( a = 45^{\circ} \) and \( b = 30^{\circ} \): \[ \sin 75^{\circ} = \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ} \] Now, we use known values: - \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \) - \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) - \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \) - \( \sin 30^{\circ} = \frac{1}{2} \) Substituting these values in: \[ \sin 75^{\circ} = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right) \] \[ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \] \[ = \frac{\sqrt{6} + \sqrt{2}}{4} \] ### (3) Calculate \( \sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ} \) This expression can be simplified using the sine addition formula: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \] where \( a = 50^{\circ} \) and \( b = 10^{\circ} \): \[ \sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ} = \sin(50^{\circ} + 10^{\circ}) = \sin 60^{\circ} \] Now, we know: \[ \sin 60^{\circ} = \frac{\sqrt{3}}{2} \] ### Final Answers 1. \( \sin 75^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \) 2. \( \sin 50^{\circ} \cos 10^{\circ} + \cos 50^{\circ} \sin 10^{\circ} = \frac{\sqrt{3}}{2} \)