QUESTION 1 1.1. Given: (i) $$ \quad \begin{array}{l} ext { (ii) } \quad A=60^{\circ} ext { and } B=30^{\circ} \ ext { (iii) } A=225^{\circ} ext { and } B=50^{\circ} \ ext { Use a calculator to evalunte each of the following: } \ ext { (a) } \cos (A-B) \ ext { (b) } \cos A-\cos B \ ext { (c) } \cos A \cos B+\sin A \cdot \sin B \ 1.2 ext { Compare the values of } \cos (A-B) ext { with the values of } \cos A-\cos B ext { and } \ \cos A \cos B+\sin A \cdot \sin B \ 1.3 ext { Using your } \operatorname{comparison} ext { in Question } 12, ext { what gencral conclusion can be made regarding } \ \cos (A-B), \cos A-\cos B ext { and } \cos A \cos B+\sin A \sin B ext { for any values of } A ext { and } B ext { ? }\end{array} $$.

Question

QUESTION 1 1.1. Given: (i) $$ \quad \begin{array}{l}	ext { (ii) } \quad A=60^{\circ} 	ext { and } B=30^{\circ} \ 	ext { (iii) } A=225^{\circ} 	ext { and } B=50^{\circ} \ 	ext { Use a calculator to evalunte each of the following: } \ 	ext { (a) } \cos (A-B) \ 	ext { (b) } \cos A-\cos B \ 	ext { (c) } \cos A \cos B+\sin A \cdot \sin B \ 1.2 	ext { Compare the values of } \cos (A-B) 	ext { with the values of } \cos A-\cos B 	ext { and } \ \cos A \cos B+\sin A \cdot \sin B \ 1.3 	ext { Using your } \operatorname{comparison} 	ext { in Question } 12, 	ext { what gencral conclusion can be made regarding } \ \cos (A-B), \cos A-\cos B 	ext { and } \cos A \cos B+\sin A \sin B 	ext { for any values of } A 	ext { and } B 	ext { ? }\end{array} $$.

UpStudy AI · Accepted Answer

Let's tackle **Question 1** step by step, evaluating each part for the given angles and drawing general conclusions based on the results.

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## **1.1 Evaluations Using a Calculator**

### **Given:**
- **Case (ii):** $$ A = 60^{\circ} $$ and $$ B = 30^{\circ} $$
- **Case (iii):** $$ A = 225^{\circ} $$ and $$ B = 50^{\circ} $$

We'll evaluate the following expressions for both cases:

1. **(a) $$ \cos(A - B) $$**
2. **(b) $$ \cos A - \cos B $$**
3. **(c) $$ \cos A \cos B + \sin A \sin B $$**

---

### **Case (ii): $$ A = 60^{\circ} $$, $$ B = 30^{\circ} $$**

1. **(a) $$ \cos(A - B) = \cos(60^{\circ} - 30^{\circ}) = \cos(30^{\circ}) $$**
   $$
   \cos(30^{\circ}) \approx 0.8660
   $$

2. **(b) $$ \cos A - \cos B = \cos(60^{\circ}) - \cos(30^{\circ}) $$**
   $$
   \cos(60^{\circ}) = 0.5 \
   \cos(30^{\circ}) \approx 0.8660 \
   \cos A - \cos B = 0.5 - 0.8660 \approx -0.3660
   $$

3. **(c) $$ \cos A \cos B + \sin A \sin B = \cos(60^{\circ}) \cos(30^{\circ}) + \sin(60^{\circ}) \sin(30^{\circ}) $$**
   $$
   \cos(60^{\circ}) = 0.5 \
   \cos(30^{\circ}) \approx 0.8660 \
   \sin(60^{\circ}) \approx 0.8660 \
   \sin(30^{\circ}) = 0.5 \
   \cos A \cos B + \sin A \sin B = (0.5)(0.8660) + (0.8660)(0.5) \
   = 0.4330 + 0.4330 = 0.8660
   $$

---

### **Case (iii): $$ A = 225^{\circ} $$, $$ B = 50^{\circ} $$**

1. **(a) $$ \cos(A - B) = \cos(225^{\circ} - 50^{\circ}) = \cos(175^{\circ}) $$**
   $$
   \cos(175^{\circ}) \approx -0.9962
   $$

2. **(b) $$ \cos A - \cos B = \cos(225^{\circ}) - \cos(50^{\circ}) $$**
   $$
   \cos(225^{\circ}) = -\frac{\sqrt{2}}{2} \approx -0.7071 \
   \cos(50^{\circ}) \approx 0.6428 \
   \cos A - \cos B = -0.7071 - 0.6428 \approx -1.3499
   $$

3. **(c) $$ \cos A \cos B + \sin A \sin B = \cos(225^{\circ}) \cos(50^{\circ}) + \sin(225^{\circ}) \sin(50^{\circ}) $$**
   $$
   \cos(225^{\circ}) \approx -0.7071 \
   \cos(50^{\circ}) \approx 0.6428 \
   \sin(225^{\circ}) \approx -0.7071 \
   \sin(50^{\circ}) \approx 0.7660 \
   \cos A \cos B + \sin A \sin B = (-0.7071)(0.6428) + (-0.7071)(0.7660) \
   \approx -0.4546 - 0.5412 = -0.9958
   $$

---

## **1.2 Comparison of Values**

Let's compile the results for both cases:

| Expression                        | Case (ii): $$ A = 60^{\circ} $$, $$ B = 30^{\circ} $$ | Case (iii): $$ A = 225^{\circ} $$, $$ B = 50^{\circ} $$ |
|-----------------------------------|--------------------------------------------------------|----------------------------------------------------------|
| **(a) $$ \cos(A - B) $$**        | $$ \approx 0.8660 $$                                   | $$ \approx -0.9962 $$                                     |
| **(b) $$ \cos A - \cos B $$**    | $$ \approx -0.3660 $$                                  | $$ \approx -1.3499 $$                                     |
| **(c) $$ \cos A \cos B + \sin A \sin B $$** | $$ \approx 0.8660 $$                          | $$ \approx -0.9958 $$                                     |

**Observations:**
- **Expression (a)** and **Expression (c)** yield **identical** results in both cases.
- **Expression (b)** yields different results and does not match **Expressions (a)** or **(c)**.

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## **1.3 General Conclusion**

Based on the comparisons:

- **$$ \cos(A - B) $$** is **always equal** to **$$ \cos A \cos B + \sin A \sin B $$** for any angles $$ A $$ and $$ B $$. This is a fundamental trigonometric identity.

- **$$ \cos A - \cos B $$** does **not** generally equal **$$ \cos(A - B) $$** or **$$ \cos A \cos B + \sin A \sin B $$**. It yields a different value unless specific conditions are met where the expressions coincidentally align.

**Therefore, the general conclusion is:**

$$
\boxed{\cos(A - B) = \cos A \cos B + \sin A \sin B \quad \text{for any angles } A \text{ and } B.}
$$

This identity underscores the relationship between the cosine of a difference of angles and the product of their sines and cosines. It distinguishes from the expression $$ \cos A - \cos B $$, which represents a different trigonometric relationship.
**General Conclusion:** $$ \cos(A - B) = \cos A \cos B + \sin A \sin B \quad \text{for any angles } A \text{ and } B. $$ This identity shows that the cosine of the difference between two angles is equal to the product of their cosines plus the product of their sines.

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