1.3 Determine, without using a calculator, the value of the following in terms of $$ p $$, if $$ \sin 43^{\circ}=p $$ $$ 1.3 .1 \cos 133^{\circ} $$ $$ 1.3 .2 an \left(-43^{\circ} ight) $$

Question

1.3 Determine, without using a calculator, the value of the following in terms of $$ p $$, if $$ \sin 43^{\circ}=p $$ $$ 1.3 .1 \cos 133^{\circ} $$ $$ 1.3 .2 	an \left(-43^{\circ}ight) $$

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Let's solve each part step by step, given that $$ \sin 43^{\circ} = p $$.

### 1.3.1 $$\cos 133^{\circ}$$

To express $$\cos 133^{\circ}$$ in terms of $$ p $$:

1. **Express $$133^{\circ}$$ in terms of a related angle:**
   $$
   133^{\circ} = 90^{\circ} + 43^{\circ}
   $$
   
2. **Use the cosine of a sum identity:**
   $$
   \cos(90^{\circ} + \theta) = -\sin \theta
   $$
   
3. **Apply the identity:**
   $$
   \cos 133^{\circ} = \cos(90^{\circ} + 43^{\circ}) = -\sin 43^{\circ} = -p
   $$

**Answer:**
$$
\cos 133^{\circ} = -p
$$

---

### 1.3.2 $$\tan(-43^{\circ})$$

To express $$\tan(-43^{\circ})$$ in terms of $$ p $$:

1. **Use the odd property of the tangent function:**
   $$
   \tan(-\theta) = -\tan \theta
   $$
   Thus,
   $$
   \tan(-43^{\circ}) = -\tan 43^{\circ}
   $$
   
2. **Express $$\tan 43^{\circ}$$ using sine and cosine:**
   $$
   \tan 43^{\circ} = \frac{\sin 43^{\circ}}{\cos 43^{\circ}} = \frac{p}{\cos 43^{\circ}}
   $$
   
3. **Find $$\cos 43^{\circ}$$ in terms of $$ p $$:**
   Using the Pythagorean identity,
   $$
   \sin^2 \theta + \cos^2 \theta = 1 \Rightarrow \cos 43^{\circ} = \sqrt{1 - \sin^2 43^{\circ}} = \sqrt{1 - p^2}
   $$
   
4. **Substitute back into the expression for $$\tan 43^{\circ}$$:**
   $$
   \tan 43^{\circ} = \frac{p}{\sqrt{1 - p^2}}
   $$
   
5. **Finally, express $$\tan(-43^{\circ})$$:**
   $$
   \tan(-43^{\circ}) = -\frac{p}{\sqrt{1 - p^2}}
   $$

**Answer:**
$$
\tan(-43^{\circ}) = -\frac{p}{\sqrt{1 - p^2}}
$$

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### **Summary of Answers**

1. **1.3.1** $$ \cos 133^{\circ} = -p $$

2. **1.3.2** $$ \tan(-43^{\circ}) = -\dfrac{p}{\sqrt{1 - p^2}} $$
$$ \cos 133^{\circ} = -p \quad \text{and} \quad \tan(-43^{\circ}) = -\frac{p}{\sqrt{1 - p^2}} $$

1.3 Determine, without using a calculator, the value of the following in terms of \( p \), if \( \sin 43^{\circ}=p \) \( 1.3 .1 \cos 133^{\circ} \) \( 1.3 .2 \tan \left(-43^{\circ}\right) \)

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