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) \( \cos A \cdot \cos B+\sin A \cdot \sin B \) \( 1.2 \quad \) 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 ., 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 \( B \) ?

Certainly! Let's systematically address each part of your problem. ## **1.1. Evaluations** We'll evaluate the following expressions for each given pair of angles \( A \) and \( B \): - **(a)** \( \cos (A - B) \) - **(b)** \( \cos A - \cos B \) - **(c)** \( \cos A \cdot \cos B + \sin A \cdot \sin B \) We'll use a calculator to find numerical values for each expression. ### **Case (i): \( A = 60^{\circ} \) and \( B = 30^{\circ} \)** 1. **(a) \( \cos (A - B) = \cos (60^{\circ} - 30^{\circ}) = \cos 30^{\circ} \)** \[ \cos 30^{\circ} = \frac{\sqrt{3}}{2} \approx 0.8660 \] 2. **(b) \( \cos A - \cos B = \cos 60^{\circ} - \cos 30^{\circ} \)** \[ \cos 60^{\circ} = 0.5 \\ \cos 30^{\circ} \approx 0.8660 \\ \cos 60^{\circ} - \cos 30^{\circ} = 0.5 - 0.8660 = -0.3660 \] 3. **(c) \( \cos A \cdot \cos B + \sin A \cdot \sin B = \cos 60^{\circ} \cdot \cos 30^{\circ} + \sin 60^{\circ} \cdot \sin 30^{\circ} \)** \[ \cos 60^{\circ} = 0.5 \\ \cos 30^{\circ} \approx 0.8660 \\ \sin 60^{\circ} \approx 0.8660 \\ \sin 30^{\circ} = 0.5 \\ \cos 60^{\circ} \cdot \cos 30^{\circ} \approx 0.5 \times 0.8660 = 0.4330 \\ \sin 60^{\circ} \cdot \sin 30^{\circ} \approx 0.8660 \times 0.5 = 0.4330 \\ \text{Sum} = 0.4330 + 0.4330 = 0.8660 \] ### **Case (ii): \( A = 110^{\circ} \) and \( B = 50^{\circ} \)** 1. **(a) \( \cos (A - B) = \cos (110^{\circ} - 50^{\circ}) = \cos 60^{\circ} \)** \[ \cos 60^{\circ} = 0.5 \] 2. **(b) \( \cos A - \cos B = \cos 110^{\circ} - \cos 50^{\circ} \)** \[ \cos 110^{\circ} \approx -0.3420 \\ \cos 50^{\circ} \approx 0.6428 \\ \cos 110^{\circ} - \cos 50^{\circ} \approx -0.3420 - 0.6428 = -0.9848 \] 3. **(c) \( \cos A \cdot \cos B + \sin A \cdot \sin B = \cos 110^{\circ} \cdot \cos 50^{\circ} + \sin 110^{\circ} \cdot \sin 50^{\circ} \)** \[ \cos 110^{\circ} \approx -0.3420 \\ \cos 50^{\circ} \approx 0.6428 \\ \sin 110^{\circ} \approx 0.9397 \\ \sin 50^{\circ} \approx 0.7660 \\ \cos 110^{\circ} \cdot \cos 50^{\circ} \approx -0.3420 \times 0.6428 \approx -0.2197 \\ \sin 110^{\circ} \cdot \sin 50^{\circ} \approx 0.9397 \times 0.7660 \approx 0.7200 \\ \text{Sum} = -0.2197 + 0.7200 \approx 0.5003 \approx 0.5 \] ### **Case (iii): \( A = 225^{\circ} \) and \( B = 135^{\circ} \)** 1. **(a) \( \cos (A - B) = \cos (225^{\circ} - 135^{\circ}) = \cos 90^{\circ} \)** \[ \cos 90^{\circ} = 0 \] 2. **(b) \( \cos A - \cos B = \cos 225^{\circ} - \cos 135^{\circ} \)** \[ \cos 225^{\circ} \approx -0.7071 \\ \cos 135^{\circ} \approx -0.7071 \\ \cos 225^{\circ} - \cos 135^{\circ} \approx -0.7071 - (-0.7071) = 0 \] 3. **(c) \( \cos A \cdot \cos B + \sin A \cdot \sin B = \cos 225^{\circ} \cdot \cos 135^{\circ} + \sin 225^{\circ} \cdot \sin 135^{\circ} \)** \[ \cos 225^{\circ} \approx -0.7071 \\ \cos 135^{\circ} \approx -0.7071 \\ \sin 225^{\circ} \approx -0.7071 \\ \sin 135^{\circ} \approx 0.7071 \\ \cos 225^{\circ} \cdot \cos 135^{\circ} \approx (-0.7071) \times (-0.7071) = 0.5 \\ \sin 225^{\circ} \cdot \sin 135^{\circ} \approx (-0.7071) \times 0.7071 = -0.5 \\ \text{Sum} = 0.5 + (-0.5) = 0 \] ## **1.2. Comparison of Values** Let's tabulate the results for clarity: | 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 | ### **Observations:** 1. **\( \cos (A - B) \) and \( \cos A \cdot \cos B + \sin A \cdot \sin B \):** - **Case (i):** Both equal to approximately 0.8660 - **Case (ii):** Both equal to approximately 0.5 - **Case (iii):** Both equal to 0 2. **\( \cos A - \cos B \):** - **Case (i):** -0.3660 - **Case (ii):** -0.9848 - **Case (iii):** 0 **Conclusion from Comparison:** - In all cases, **\( \cos (A - B) \)** is **equal** to **\( \cos A \cdot \cos B + \sin A \cdot \sin B \)**. - **\( \cos A - \cos B \)** does **not** generally equal **\( \cos (A - B) \)** or **\( \cos A \cdot \cos B + \sin A \cdot \sin B \)**, except in specific instances (e.g., Case (iii)). ## **1.3. General Conclusion** Based on the comparisons: - **Identity Confirmation:** \[ \cos (A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \] This is a fundamental trigonometric identity and holds true **for any angles \( A \) and \( B \)**. - **Difference of Cosines (\( \cos A - \cos B \)):** The expression \( \cos A - \cos B \) does **not** generally equal \( \cos (A - B) \) or \( \cos A \cdot \cos B + \sin A \cdot \sin B \). Its equality to other expressions depends on specific values of \( A \) and \( B \), as seen in the cases above. **Final Conclusion:** For any angles \( A \) and \( B \), \[ \boxed{\cos (A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B} \] This identity consistently holds, while \( \cos A - \cos B \) does not generally equate to \( \cos (A - B) \) or \( \cos A \cdot \cos B + \sin A \cdot \sin B \).