Given: (i) \( A=60^{\circ} \) and \( B=30^{\circ} \) (ii) \( A=110^{\circ} \) and \( B=50^{\circ} \) (iti) \( A=225^{\circ} \) 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 evaluate the expressions with the given angles, let's work through each pair of angles step by step: ### For \( A = 60^{\circ} \) and \( B = 30^{\circ} \): 1. **(a)** \( \cos (A-B) = \cos(60^{\circ} - 30^{\circ}) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \approx 0.866 \) 2. **(b)** \( \cos A - \cos B = \cos(60^{\circ}) - \cos(30^{\circ}) = \frac{1}{2} - \frac{\sqrt{3}}{2} \approx -0.366 \) 3. **(c)** \( \cos A \cos B + \sin A \sin B = \cos(60^{\circ}) \cos(30^{\circ}) + \sin(60^{\circ}) \sin(30^{\circ}) \) = \( \frac{1}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \approx 0.866 \) ### For \( A = 110^{\circ} \) and \( B = 50^{\circ} \): 1. **(a)** \( \cos (A-B) = \cos(110^{\circ} - 50^{\circ}) = \cos(60^{\circ}) = \frac{1}{2} = 0.5 \) 2. **(b)** \( \cos A - \cos B = \cos(110^{\circ}) - \cos(50^{\circ}) \approx -0.342 - 0.643 \approx -0.985 \) 3. **(c)** \( \cos A \cos B + \sin A \sin B \) = \( \cos(110^{\circ}) \cos(50^{\circ}) + \sin(110^{\circ}) \sin(50^{\circ}) \) = \( (-0.342)(0.643) + (0.940)(0.766) \approx -0.220 + 0.719 = 0.499 \approx 0.5 \) ### For \( A = 225^{\circ} \) and \( B = 135^{\circ} \): 1. **(a)** \( \cos (A-B) = \cos(225^{\circ} - 135^{\circ}) = \cos(90^{\circ}) = 0 \) 2. **(b)** \( \cos A - \cos B = \cos(225^{\circ}) - \cos(135^{\circ}) = -\frac{\sqrt{2}}{2} - (-\frac{\sqrt{2}}{2}) = 0 \) 3. **(c)** \( \cos A \cos B + \sin A \sin B \) = \( \cos(225^{\circ}) \cos(135^{\circ}) + \sin(225^{\circ}) \sin(135^{\circ}) \) = \( (-\frac{\sqrt{2}}{2})(-\frac{\sqrt{2}}{2}) + (-\frac{\sqrt{2}}{2})(\frac{\sqrt{2}}{2}) = \frac{1}{2} - \frac{1}{2} = 0 \) ### Summary of Results: - For \( A = 60^\circ \), \( B = 30^\circ \): - (a) \( \frac{\sqrt{3}}{2} \approx 0.866 \) - (b) \( \frac{1}{2} - \frac{\sqrt{3}}{2} \approx -0.366 \) - (c) \( \frac{\sqrt{3}}{2} \approx 0.866 \) - For \( A = 110^\circ \), \( B = 50^\circ \): - (a) \( 0.5 \) - (b) \( -0.985 \) - (c) \( 0.5 \) - For \( A = 225^\circ \), \( B = 135^\circ \): - (a) \( 0 \) - (b) \( 0 \)