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

To solve these problems, we will use trigonometric identities and a calculator for the numerical evaluations. ### (i) \( A=60^{\circ} \) and \( B=30^{\circ} \) (a) \( \cos (A-B) = \cos(60^\circ - 30^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 \) (b) \( \cos A - \cos B = \cos 60^\circ - \cos 30^\circ = \frac{1}{2} - \frac{\sqrt{3}}{2} \approx -0.366 \) (c) \( \cos A \cos B + \sin A \sin B = \cos 60^\circ \cos 30^\circ + \sin 60^\circ \sin 30^\circ = \left(\frac{1}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{3}}{2} \cdot \frac{1}{2}\right) = \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \approx 0.866 \) --- ### (ii) \( A=110^{\circ} \) and \( B=50^{\circ} \) (a) \( \cos (A-B) = \cos(110^\circ - 50^\circ) = \cos(60^\circ) = \frac{1}{2} = 0.5 \) (b) \( \cos A - \cos B = \cos 110^\circ - \cos 50^\circ \approx -0.342 - 0.643 = -0.985 \) (c) \( \cos A \cos B + \sin A \sin B = \cos 110^\circ \cos 50^\circ + \sin 110^\circ \sin 50^\circ \approx (-0.342 \cdot 0.643) + (0.951 \cdot 0.766) \approx -0.220 + 0.728 = 0.508 \) --- ### (iii) \( A=225^{\circ} \) and \( B=135^{\circ} \) (a) \( \cos (A-B) = \cos(225^\circ - 135^\circ) = \cos(90^\circ) = 0 \) (b) \( \cos A - \cos B = \cos 225^\circ - \cos 135^\circ \approx -0.707 - (-0.707) = 0 \) (c) \( \cos A \cos B + \sin A \sin B = \cos 225^\circ \cos 135^\circ + \sin 225^\circ \sin 135^\circ \approx (-0.707)(-0.707) + (-0.707)(0.707) = 0.5 - 0.5 = 0 \) --- So the results for each case are: 1. **(i)** \(\cos (A-B) \approx 0.866\), \(\cos A - \cos B \approx -0.366\), \(\cos A \cos B + \sin A \sin B \approx 0.866\) 2. **(ii)** \(\cos (A-B) = 0.5\), \(\cos A - \cos B \approx -0.985\), \(\cos A \cos B + \sin A \sin B \approx 0.508\) 3. **(iii)** \(\cos (A-B) = 0\), \(\cos A - \cos B = 0\), \(\cos A \cos B + \sin A \sin B = 0\)