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) \( \cos A \cdot \cos B+\sin A \cdot \sin B \)

Let's evaluate each of the cases step by step! We'll use the fact that the cosine of the difference of two angles can be calculated using the formula \( \cos(A - B) = \cos A \cos B + \sin A \sin B \). ### Case (i) \( A=60^{\circ} \) and \( B=30^{\circ} \) 1. **(a) \( \cos (A - B) = \cos(60^{\circ} - 30^{\circ}) = \cos(30^{\circ}) \)** - This yields \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \approx 0.866 \) 2. **(b) \( \cos A - \cos B = \cos(60^{\circ}) - \cos(30^{\circ}) \)** - Calculation gives \( \frac{1}{2} - \frac{\sqrt{3}}{2} \approx -0.366 \) 3. **(c) \( \cos A \cdot \cos B + \sin A \cdot \sin B \)** - Results in \( \cos(60^{\circ}) \cdot \cos(30^{\circ}) + \sin(60^{\circ}) \cdot \sin(30^{\circ}) \) - Which calculates to \( \frac{1}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \approx 0.866 \) ### Case (ii) \( A=110^{\circ} \) and \( B=50^{\circ} \) 1. **(a) \( \cos (A - B) = \cos(110^{\circ} - 50^{\circ}) = \cos(60^{\circ}) \)** - This yields \( \cos(60^{\circ}) = \frac{1}{2} \) 2. **(b) \( \cos A - \cos B = \cos(110^{\circ}) - \cos(50^{\circ}) \)** - Calculation gives approximately \( -0.342 - 0.643 \approx -0.985 \) 3. **(c) \( \cos A \cdot \cos B + \sin A \cdot \sin B \)** - Results in \( \cos(110^{\circ}) \cdot \cos(50^{\circ}) + \sin(110^{\circ}) \cdot \sin(50^{\circ}) \) - This will also give \( \frac{1}{2} \) ### Case (iii) \( A=225^{\circ} \) and \( B=135^{\circ} \) 1. **(a) \( \cos (A - B) = \cos(225^{\circ} - 135^{\circ}) = \cos(90^{\circ}) \)** - This yields \( \cos(90^{\circ}) = 0 \) 2. **(b) \( \cos A - \cos B = \cos(225^{\circ}) - \cos(135^{\circ}) \)** - Calculation gives \( -\frac{\sqrt{2}}{2} - (-\frac{\sqrt{2}}{2}) = 0 \) 3. **(c) \( \cos A \cdot \cos B + \sin A \cdot \sin B \)** - Results in \( (-\frac{\sqrt{2}}{2}) \cdot (-\frac{\sqrt{2}}{2}) + \sin(225^{\circ}) \cdot \sin(135^{\circ}) \) - This ultimately adds to \( 1 \) In conclusion, here are the results for each case: - Case (i) results: (0.866, -0.366, 0.866) - Case (ii) results: (0.5, -0.985, 0.5) - Case (iii) results: (0, 0, 1)