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 Compare the values of \( \cos (A-B) \) with the values of \( \cos A-\cos B \) and \( \cos A \cdot \cos B+\sin A \cdot \sin B \) \( 1.3 \quad \) Using your \( \operatorname{comparison} \) in Question \( 1.2 \cdot \), what general \( \operatorname{conclusion} \operatorname{con} \) 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 tackle each part of the problem systematically. ## **1.1 Evaluations Using a Calculator** Given three pairs of angles: 1. **(i)** \( A = 60^\circ \) and \( B = 30^\circ \) 2. **(ii)** \( A = 110^\circ \) and \( B = 50^\circ \) 3. **(iii)** \( A = 225^\circ \) and \( B = 135^\circ \) We'll evaluate the following expressions for each pair: - **(a)** \( \cos(A - B) \) - **(b)** \( \cos A - \cos B \) - **(c)** \( \cos A \cdot \cos B + \sin A \cdot \sin B \) ### **(a) Evaluating \( \cos(A - B) \)** Using the angle difference, calculate \( A - B \) and then find the cosine: 1. **(i)** \[ A - B = 60^\circ - 30^\circ = 30^\circ \\ \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660 \] 2. **(ii)** \[ A - B = 110^\circ - 50^\circ = 60^\circ \\ \cos(60^\circ) = 0.5 \] 3. **(iii)** \[ A - B = 225^\circ - 135^\circ = 90^\circ \\ \cos(90^\circ) = 0 \] ### **(b) Evaluating \( \cos A - \cos B \)** Calculate the cosine of each angle and then subtract: 1. **(i)** \[ \cos(60^\circ) = 0.5 \\ \cos(30^\circ) \approx 0.8660 \\ \cos A - \cos B = 0.5 - 0.8660 = -0.3660 \] 2. **(ii)** \[ \cos(110^\circ) \approx -0.3420 \quad (\text{since } 110^\circ \text{ is in the second quadrant}) \\ \cos(50^\circ) \approx 0.6428 \\ \cos A - \cos B = -0.3420 - 0.6428 = -0.9848 \] 3. **(iii)** \[ \cos(225^\circ) \approx -0.7071 \quad (\text{since } 225^\circ \text{ is in the third quadrant}) \\ \cos(135^\circ) \approx -0.7071 \quad (\text{since } 135^\circ \text{ is in the second quadrant}) \\ \cos A - \cos B = -0.7071 - (-0.7071) = 0 \] ### **(c) Evaluating \( \cos A \cdot \cos B + \sin A \cdot \sin B \)** This expression is derived from the cosine of a difference identity, \( \cos(A - B) = \cos A \cos B + \sin A \sin B \). Let's compute it to verify: 1. **(i)** \[ \cos(60^\circ) = 0.5, \quad \cos(30^\circ) \approx 0.8660 \\ \sin(60^\circ) \approx 0.8660, \quad \sin(30^\circ) = 0.5 \\ \cos A \cdot \cos B + \sin A \cdot \sin B = (0.5)(0.8660) + (0.8660)(0.5) = 0.4330 + 0.4330 = 0.8660 \] 2. **(ii)** \[ \cos(110^\circ) \approx -0.3420, \quad \cos(50^\circ) \approx 0.6428 \\ \sin(110^\circ) \approx 0.9397, \quad \sin(50^\circ) \approx 0.7660 \\ \cos A \cdot \cos B + \sin A \cdot \sin B = (-0.3420)(0.6428) + (0.9397)(0.7660) \approx -0.2197 + 0.7205 = 0.5008 \approx 0.5 \] 3. **(iii)** \[ \cos(225^\circ) \approx -0.7071, \quad \cos(135^\circ) \approx -0.7071 \\ \sin(225^\circ) \approx -0.7071, \quad \sin(135^\circ) \approx 0.7071 \\ \cos A \cdot \cos B + \sin A \cdot \sin B = (-0.7071)(-0.7071) + (-0.7071)(0.7071) \approx 0.5 - 0.5 = 0 \] ## **1.2 Comparison of Values** Let's summarize the results: | Pair | \( \cos(A - B) \) | \( \cos A - \cos B \) | \( \cos A \cdot \cos B + \sin A \cdot \sin B \) | |------|-------------------|------------------------|-----------------------------------------------| | (i) | 0.8660 | -0.3660 | 0.8660 | | (ii) | 0.5 | -0.9848 | 0.5 | | (iii)| 0 | 0 | 0 | **Observations:** - **\( \cos(A - B) \)** and **\( \cos A \cdot \cos B + \sin A \cdot \sin B \)** yield the **same values** for all angle pairs. - **\( \cos A - \cos B \)** generally **differs** from the other two expressions, except in specific cases (e.g., when \( A = B \) or when their cosines cancel out). ## **1.3 General Conclusion** Based on the comparisons: - The expression \( \cos(A - B) \) **always equals** \( \cos A \cdot \cos B + \sin A \cdot \sin B \). This is a fundamental trigonometric identity known as the **cosine of a difference identity**. - The expression \( \cos A - \cos B \) is **generally distinct** from \( \cos(A - B) \) and \( \cos A \cdot \cos B + \sin A \cdot \sin B \), except in particular cases where the values align due to specific angle relationships. **Therefore, for any angles \( A \) and \( B \):** \[ \cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \] While \( \cos(A - B) \) and \( \cos A \cdot \cos B + \sin A \cdot \sin B \) are identically equal, \( \cos A - \cos B \) is generally not equal to these expressions.