UESTION 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: \( \begin{array}{l}\text { (a) } \cos (A-B) \\ \text { (b) } \cos A-\cos B \\ \text { (c) } \cos A \cdot \cos B+\sin A \cdot \sin B \\ 2\end{array} \) Compare the values of \( \cos (A-B) \) with the values of \( \cos A-\cos B \) and \( \cos A \cdot \cos B+\sin A \cdot \sin B \) Using your \( \operatorname{comparison} \) in Question \( 1.2 \cdot \), what general \( \operatorname{conclusion~can~be~made~regarding~} \) \( \cos (A-B), \cos A-\cos B \) and \( \cos A \cdot \cos B+\sin A \cdot \sin B \) for any values of \( A \) and \( B \) ?

Let's evaluate each part of the question step-by-step for the given angle pairs and then draw a general conclusion. ### Given Angle Pairs: 1. **Case (i):** \( A = 60^{\circ} \), \( B = 30^{\circ} \) 2. **Case (ii):** \( A = 110^{\circ} \), \( B = 50^{\circ} \) 3. **Case (iii):** \( A = 225^{\circ} \), \( B = 135^{\circ} \) ### Part (a): Evaluate \( \cos(A - B) \) Using the cosine difference identity: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \] Let's calculate for each case: 1. **Case (i):** \[ \cos(60^{\circ} - 30^{\circ}) = \cos(30^{\circ}) \approx 0.8660 \] 2. **Case (ii):** \[ \cos(110^{\circ} - 50^{\circ}) = \cos(60^{\circ}) = 0.5 \] 3. **Case (iii):** \[ \cos(225^{\circ} - 135^{\circ}) = \cos(90^{\circ}) = 0 \] ### Part (b): Evaluate \( \cos A - \cos B \) Calculate for each case: 1. **Case (i):** \[ \cos(60^{\circ}) - \cos(30^{\circ}) = 0.5 - 0.8660 = -0.3660 \] 2. **Case (ii):** \[ \cos(110^{\circ}) - \cos(50^{\circ}) \approx -0.3420 - 0.6428 = -0.9848 \] 3. **Case (iii):** \[ \cos(225^{\circ}) - \cos(135^{\circ}) \approx -0.7071 - (-0.7071) = 0 \] ### Part (c): Evaluate \( \cos A \cdot \cos B + \sin A \cdot \sin B \) Using the cosine difference identity again: \[ \cos A \cdot \cos B + \sin A \cdot \sin B = \cos(A - B) \] Thus, the values will be identical to Part (a): 1. **Case (i):** \( \approx 0.8660 \) 2. **Case (ii):** \( 0.5 \) 3. **Case (iii):** \( 0 \) ### Comparison of Values: | Case | \( \cos(A - B) \) | \( \cos A - \cos B \) | \( \cos A \cdot \cos B + \sin A \cdot \sin B \) | |------|-------------------|------------------------|-------------------------------------------------| | (i) | 0.8660 | -0.3660 | 0.8660 | | (ii) | 0.5 | -0.9848 | 0.5 | | (iii)| 0 | 0 | 0 | ### General Conclusion: From the calculations above, we observe that: - **\( \cos(A - B) \)** and **\( \cos A \cdot \cos B + \sin A \cdot \sin B \)** yield the **same** results for all cases. - **\( \cos A - \cos B \)** generally produces different values and does not align with \( \cos(A - B) \). **Therefore, the general conclusion is:** For any angles \( A \) and \( B \), \[ \cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \] However, \( \cos(A - B) \) is **not equal** to \( \cos A - \cos B \) except in specific cases where the angles result in coinciding values (as seen in Case (iii)).