1.1. Given: (i) \( A=60^{\circ} \) and \( B=30^{\circ} \) (ii) \( A=110^{\circ} \) and \( B=50^{\circ} \) (iii) \( A=225^{\circ} \) and \( B=135^{\circ} \) Use a calculator to evaluate each of the following: (a) \( \cos (A-B) \) (b) \( \cos A-\cos B \) (c) \( \cos A \cdot \cos B+\sin A \cdot \sin B \)

To solve the given angles for \( A \) and \( B \), we will evaluate the expressions for each pair of angles. ### (i) \( A=60^{\circ} \) and \( B=30^{\circ} \) (a) \( \cos (A-B) = \cos(60^{\circ} - 30^{\circ}) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \) (b) \( \cos A - \cos B = \cos(60^{\circ}) - \cos(30^{\circ}) = \frac{1}{2} - \frac{\sqrt{3}}{2} = \frac{1 - \sqrt{3}}{2} \) (c) \( \cos A \cdot \cos B + \sin A \cdot \sin B = \cos(60^{\circ}) \cdot \cos(30^{\circ}) + \sin(60^{\circ}) \cdot \sin(30^{\circ}) = \frac{1}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \) ### (ii) \( A=110^{\circ} \) and \( B=50^{\circ} \) (a) \( \cos (A-B) = \cos(110^{\circ} - 50^{\circ}) = \cos(60^{\circ}) = \frac{1}{2} \) (b) \( \cos A - \cos B = \cos(110^{\circ}) - \cos(50^{\circ}) \approx -0.342 - 0.643 \approx -0.985 \) (c) \( \cos A \cdot \cos B + \sin A \cdot \sin B = \cos(110^{\circ}) \cdot \cos(50^{\circ}) + \sin(110^{\circ}) \cdot \sin(50^{\circ}) \approx (-0.342)\cdot(0.643) + (0.940)\cdot(0.766) \approx -0.220 + 0.719 \approx 0.499 \) ### (iii) \( A=225^{\circ} \) and \( B=135^{\circ} \) (a) \( \cos (A-B) = \cos(225^{\circ} - 135^{\circ}) = \cos(90^{\circ}) = 0 \) (b) \( \cos A - \cos B = \cos(225^{\circ}) - \cos(135^{\circ}) = -\frac{\sqrt{2}}{2} - (-\frac{\sqrt{2}}{2}) = 0 \) (c) \( \cos A \cdot \cos B + \sin A \cdot \sin B = \cos(225^{\circ}) \cdot \cos(135^{\circ}) + \sin(225^{\circ}) \cdot \sin(135^{\circ}) \approx \left(-\frac{\sqrt{2}}{2}\right)\left(-\frac{\sqrt{2}}{2}\right) + \left(-\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) = \frac{1}{2} - \frac{1}{2} = 0 \) Now you have all the requested evaluations for each scenario!