Mathematics Grade 12, 3CAPS, Investigation: February 2025 INVESTIGATING COMPOUND ANGLES AND THEREFORE, DOUBLE ANGLES. QUESTION 1 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 1.2 Compare the values of cos (A-B) with the values of cos A-cos B and cos A.cos B+sin A.sin B 1.3 Using your comparison in Question 1.2., what general conclusion can be made regarding cos (A-B), cos A-cos B and cos A.cos B+sin A.sin B for any values of A and B?

Certainly! Let's work through each part of **Question 1** step by step. --- ## **Question 1.1** ### **Given:** 1. **Pair (i):** - \( A = 60^\circ \) - \( B = 30^\circ \) 2. **Pair (ii):** - \( A = 110^\circ \) - \( B = 50^\circ \) 3. **Pair (iii):** - \( A = 225^\circ \) - \( B = 135^\circ \) For each pair, we'll evaluate the following expressions using a calculator: - **(a)** \( \cos(A - B) \) - **(b)** \( \cos A - \cos B \) - **(c)** \( \cos A \cdot \cos B + \sin A \cdot \sin B \) --- ### **Pair (i): \( A = 60^\circ \), \( B = 30^\circ \)** 1. **(a) \( \cos(A - B) \):** \[ \cos(60^\circ - 30^\circ) = \cos(30^\circ) \approx 0.8660 \] 2. **(b) \( \cos A - \cos B \):** \[ \cos(60^\circ) - \cos(30^\circ) \approx 0.5000 - 0.8660 = -0.3660 \] 3. **(c) \( \cos A \cdot \cos B + \sin A \cdot \sin B \):** \[ \cos(60^\circ) \cdot \cos(30^\circ) + \sin(60^\circ) \cdot \sin(30^\circ) \approx (0.5000 \times 0.8660) + (0.8660 \times 0.5000) = 0.4330 + 0.4330 = 0.8660 \] --- ### **Pair (ii): \( A = 110^\circ \), \( B = 50^\circ \)** 1. **(a) \( \cos(A - B) \):** \[ \cos(110^\circ - 50^\circ) = \cos(60^\circ) \approx 0.5000 \] 2. **(b) \( \cos A - \cos B \):** \[ \cos(110^\circ) - \cos(50^\circ) \approx (-0.3420) - 0.6428 = -0.9848 \] 3. **(c) \( \cos A \cdot \cos B + \sin A \cdot \sin B \):** \[ \cos(110^\circ) \cdot \cos(50^\circ) + \sin(110^\circ) \cdot \sin(50^\circ) \approx (-0.3420 \times 0.6428) + (0.9397 \times 0.7660) \approx -0.2197 + 0.7202 = 0.5005 \] *(Note: Small discrepancies may occur due to rounding errors.)* --- ### **Pair (iii): \( A = 225^\circ \), \( B = 135^\circ \)** 1. **(a) \( \cos(A - B) \):** \[ \cos(225^\circ - 135^\circ) = \cos(90^\circ) = 0 \] 2. **(b) \( \cos A - \cos B \):** \[ \cos(225^\circ) - \cos(135^\circ) \approx (-0.7071) - (-0.7071) = 0 \] 3. **(c) \( \cos A \cdot \cos B + \sin A \cdot \sin B \):** \[ \cos(225^\circ) \cdot \cos(135^\circ) + \sin(225^\circ) \cdot \sin(135^\circ) \approx (-0.7071 \times -0.7071) + (-0.7071 \times 0.7071) = 0.5000 - 0.5000 = 0 \] --- ## **Summary of Results:** | **Pair** | **(a) \( \cos(A - B) \)** | **(b) \( \cos A - \cos B \)** | **(c) \( \cos A \cdot \cos B + \sin A \cdot \sin B \)** | |----------|---------------------------|-------------------------------|----------------------------------------------------------| | (i) | 0.8660 | -0.3660 | 0.8660 | | (ii) | 0.5000 | -0.9848 | 0.5005 | | (iii) | 0 | 0 | 0 | --- ## **Question 1.2** ### **Comparison of Values:** From the summary above, observe the following for each pair: 1. **Pair (i):** - \( \cos(A - B) \approx 0.8660 \) - \( \cos A - \cos B \approx -0.3660 \) - \( \cos A \cdot \cos B + \sin A \cdot \sin B \approx 0.8660 \) **Observation:** \( \cos(A - B) \) and \( \cos A \cdot \cos B + \sin A \cdot \sin B \) are equal, while \( \cos A - \cos B \) has a different value. 2. **Pair (ii):** - \( \cos(A - B) \approx 0.5000 \) - \( \cos A - \cos B \approx -0.9848 \) - \( \cos A \cdot \cos B + \sin A \cdot \sin B \approx 0.5005 \) **Observation:** \( \cos(A - B) \) and \( \cos A \cdot \cos B + \sin A \cdot \sin B \) are approximately equal, while \( \cos A - \cos B \) differs. 3. **Pair (iii):** - \( \cos(A - B) = 0 \) - \( \cos A - \cos B = 0 \) - \( \cos A \cdot \cos B + \sin A \cdot \sin B = 0 \) **Observation:** All three expressions yield the same value. --- ## **Question 1.3** ### **General Conclusion:** Based on the comparisons: - **\( \cos(A - B) \)** and **\( \cos A \cdot \cos B + \sin A \cdot \sin B \)** yield the **same** values for all given pairs of angles (i, ii, iii). - **\( \cos A - \cos B \)** does **not** consistently equal \( \cos(A - B) \) or \( \cos A \cdot \cos B + \sin A \cdot \sin B \) except in specific cases (e.g., Pair (iii)). ### **Conclusion:** For **any** values of \( A \) and \( B \): \[ \cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \] This is a fundamental trigonometric identity known as the **Cosine of a Difference Identity**. However, \( \cos(A - B) \) is **not** generally equal to \( \cos A - \cos B \). ---