QUESTION 4 4.1 Use the identity for $$ \sin (A+B)=\sin A \cos B+\cos A \sin B $$ to derive a formula for $$ \cos (A-B) $$ 4.2 Without using a calculator determine the value of: $$ \cos \left(x+20^{\circ} ight) \cos \left(x-40^{\circ} ight)+\sin \left(x+20^{\circ} ight) \sin \left(x-40^{\circ} ight) $$ 4.3 If $$ \sin 31^{\circ}=k $$, express in terms of $$ k $$ : $$ 4.3 .1 \quad \sin 211^{\circ} $$ $$ 4.3 .2 \quad \sin 62^{\circ} $$ $$ 4.3 .3 \cos 1^{\circ} $$ 4.4 Determine without the use of a calculator the value of: $$ 3-2 \cos ^{2} 22.5^{\circ} $$ 4.5 Prove that: $$ \frac{\sin heta+\sin 2 heta}{1+\cos heta+\cos 2 heta}= an heta $$

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QUESTION 4 4.1 Use the identity for $$ \sin (A+B)=\sin A \cos B+\cos A \sin B $$ to derive a formula for $$ \cos (A-B) $$ 4.2 Without using a calculator determine the value of: $$ \cos \left(x+20^{\circ}ight) \cos \left(x-40^{\circ}ight)+\sin \left(x+20^{\circ}ight) \sin \left(x-40^{\circ}ight) $$ 4.3 If $$ \sin 31^{\circ}=k $$, express in terms of $$ k $$ : $$ 4.3 .1 \quad \sin 211^{\circ} $$ $$ 4.3 .2 \quad \sin 62^{\circ} $$ $$ 4.3 .3 \cos 1^{\circ} $$ 4.4 Determine without the use of a calculator the value of: $$ 3-2 \cos ^{2} 22.5^{\circ} $$ 4.5 Prove that: $$ \frac{\sin 	heta+\sin 2 	heta}{1+\cos 	heta+\cos 2 	heta}=	an 	heta $$

UpStudy AI · Accepted Answer

Let's solve each part of the question step by step.

### 4.1 Deriving a formula for $$ \cos (A-B) $$

We start with the identity for $$ \sin(A + B) $$:
$$
\sin(A + B) = \sin A \cos B + \cos A \sin B
$$

To find $$ \cos(A - B) $$, we can use the identity for $$ \cos(A + B) $$:
$$
\cos(A + B) = \cos A \cos B - \sin A \sin B
$$

Now, substituting $$ B $$ with $$ -B $$:
$$
\cos(A - B) = \cos A \cos(-B) - \sin A \sin(-B
$$
Since $$ \cos(-B) = \cos B $$ and $$ \sin(-B) = -\sin B $$, we have:
$$
\cos(A - B) = \cos A \cos B + \sin A \sin B
$$

Thus, the derived formula for $$ \cos(A - B) $$ is:
$$
\cos(A - B) = \cos A \cos B + \sin A \sin B
$$

### 4.2 Evaluating $$ \cos \left(x+20^{\circ}\right) \cos \left(x-40^{\circ}\right)+\sin \left(x+20^{\circ}\right) \sin \left(x-40^{\circ}\right) $$

Using the cosine addition formula:
$$
\cos(A - B) = \cos A \cos B + \sin A \sin B
$$
we can rewrite the expression as:
$$
\cos \left((x + 20^{\circ}) - (x - 40^{\circ})\right) = \cos(20^{\circ} + 40^{\circ}) = \cos(60^{\circ})
$$

Since $$ \cos(60^{\circ}) = \frac{1}{2} $$, the value is:
$$
\frac{1}{2}
$$

### 4.3 Expressing in terms of $$ k $$

Given $$ \sin 31^{\circ} = k $$:

#### 4.3.1 Finding $$ \sin 211^{\circ} $$

Using the identity $$ \sin(180^{\circ} + \theta) = -\sin \theta $$:
$$
\sin 211^{\circ} = \sin(180^{\circ} + 31^{\circ}) = -\sin 31^{\circ} = -k
$$

#### 4.3.2 Finding $$ \sin 62^{\circ} $$

Using the identity $$ \sin(90^{\circ} - \theta) = \cos \theta $$:
$$
\sin 62^{\circ} = \sin(90^{\circ} - 28^{\circ}) = \cos 28^{\circ}
$$
Using $$ \cos 28^{\circ} = \sin(90^{\circ} - 28^{\circ}) = \sin 62^{\circ} $$, we can express it in terms of $$ k $$ using $$ \sin^2 31^{\circ} + \cos^2 31^{\circ} = 1 $$:
$$
\cos 28^{\circ} = \sqrt{1 - k^2}
$$

#### 4.3.3 Finding $$ \cos 1^{\circ} $$

Using the identity $$ \cos(90^{\circ} - \theta) = \sin \theta $$:
$$
\cos 1^{\circ} = \sin(90^{\circ} - 1^{\circ}) = \sin 89^{\circ} \approx 1
$$

### 4.4 Evaluating $$ 3 - 2 \cos^2 22.5^{\circ} $$

Using the double angle identity:
$$
\cos 2\theta = 2\cos^2 \theta - 1 \implies 2\cos^2 \theta = 1 + \cos 2\theta
$$
Thus:
$$
3 - 2\cos^2 22.5^{\circ} = 3 - (1 + \cos 45^{\circ}) = 2 - \frac{\sqrt{2}}{2}
$$

### 4.5 Proving $$ \frac{\sin \theta + \sin 2\theta}{1 + \cos \theta + \cos 2\theta} = \tan \theta $$

Using the double angle identities:
$$
\sin 2\theta = 2\sin \theta \cos \theta \quad \text{and} \quad \cos 2\theta = 2\cos^2 \theta - 1
$$
Substituting these into the left side:
$$
\frac{\sin \theta + 2\sin \theta \cos \theta}{1 + \cos \theta + (2\cos^2 \theta - 1)} = \frac{\sin \theta(1 + 2\cos \theta)}{2\cos^2 \theta + \cos \theta}
$$
This simplifies to:
$$
\frac{\sin \theta(1 + 2\cos \theta)}{\cos \theta(2\cos \theta + 1)} = \tan \theta
$$

Thus, the proof is complete.
1. **Deriving $$ \cos(A - B) $$:** $$ \cos(A - B) = \cos A \cos B + \sin A \sin B $$ 2. **Evaluating the Expression:** $$ \cos \left(x+20^{\circ}\right) \cos \left(x-40^{\circ}\right) + \sin \left(x+20^{\circ}\right) \sin \left(x-40^{\circ}\right) = \frac{1}{2} $$ 3. **Expressing in Terms of $$ k $$:** - $$ \sin 211^{\circ} = -k $$ - $$ \sin 62^{\circ} = \sqrt{1 - k^2} $$ - $$ \cos 1^{\circ} \approx 1 $$ 4. **Evaluating $$ 3 - 2 \cos^2 22.5^{\circ} $$:** $$ 3 - 2 \cos^2 22.5^{\circ} = 2 - \frac{\sqrt{2}}{2} $$ 5. **Proving the Identity:** $$ \frac{\sin \theta + \sin 2\theta}{1 + \cos \theta + \cos 2\theta} = \tan \theta $$ (Proof steps as above)

QUESTION 4
4.1 Use the identity for
to derive a formula for
4.2 Without using a calculator determine the value of:

4.3 If , express in terms of :

4.4 Determine without the use of a calculator the value of:
4.5 Prove that:

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QUESTION 4 4.1 Use the identity for to derive a formula for 4.2 Without using a calculator determine the value of: 4.3 If , express in terms of : 4.4 Determine without the use of a calculator the value of: 4.5 Prove that: