(c) If $$ 00038^{\circ} m p $$, write the following in terms of $$ p s_{ ext {; }} $$. (1) $$ \cos 322^{\circ} $$ (2) $$ t \ln 52^{\circ} $$ (J) $$ \sin 232^{\circ} $$ (4) $$ t an 38^{\circ} $$ (5) sha $$ 142^{\circ} $$ (d) If $$ an 200^{\circ}=k $$, write the following la terms of $$ E $$ : (I) $$ \quad an 20^{\circ} $$ (2) $$ an 340^{\circ} $$. (J) $$ an \left(-160^{\circ} ight) $$ (4) $$ \sin 200^{\circ} $$ (5) $$ \cos 20^{\circ} $$ : (b) If $$ \cos 110^{\circ}=m $$, write the following in terms of mi (I) $$ \cos 70^{\circ} $$ (2) $$ \cos 290^{\circ} $$ (3) $$ \quad 160^{\circ} $$ (4). $$ \cos 20^{\circ} $$ (5) ) $$ 110^{+1} $$

Question

(c) If $$ 00038^{\circ} m p $$, write the following in terms of $$ p s_{	ext {; }} $$. (1) $$ \cos 322^{\circ} $$ (2) $$ t \ln 52^{\circ} $$ (J) $$ \sin 232^{\circ} $$ (4) $$ t 	an 38^{\circ} $$ (5) sha $$ 142^{\circ} $$ (d) If $$ 	an 200^{\circ}=k $$, write the following la terms of $$ E $$ : (I) $$ \quad 	an 20^{\circ} $$ (2) $$ 	an 340^{\circ} $$. (J) $$ 	an \left(-160^{\circ}ight) $$ (4) $$ \sin 200^{\circ} $$ (5) $$ \cos 20^{\circ} $$ : (b) If $$ \cos 110^{\circ}=m $$, write the following in terms of mi (I) $$ \cos 70^{\circ} $$ (2) $$ \cos 290^{\circ} $$ (3) $$ \quad 160^{\circ} $$ (4). $$ \cos 20^{\circ} $$ (5) ) $$ 110^{+1} $$

UpStudy AI · Accepted Answer

Let's break down the problem step by step, focusing on each part separately.

### Part (c)
We need to express the given trigonometric functions in terms of $$ p $$.

1. **For $$ \cos 322^{\circ} $$**:
   $$
   \cos 322^{\circ} = \cos(360^{\circ} - 38^{\circ}) = \cos 38^{\circ} = p
   $$

2. **For $$ t \ln 52^{\circ} $$**:
   This expression does not directly relate to $$ p $$ or any trigonometric function, so we leave it as is:
   $$
   t \ln 52^{\circ}
   $$

3. **For $$ \sin 232^{\circ} $$**:
   $$
   \sin 232^{\circ} = \sin(180^{\circ} + 52^{\circ}) = -\sin 52^{\circ}
   $$

4. **For $$ t \tan 38^{\circ} $$**:
   $$
   \tan 38^{\circ} = \frac{\sin 38^{\circ}}{\cos 38^{\circ}} = \frac{p}{\sqrt{1 - p^2}}
   $$
   Thus, $$ t \tan 38^{\circ} = t \cdot \frac{p}{\sqrt{1 - p^2}} $$.

5. **For $$ \sha 142^{\circ} $$**:
   This seems to be a typo. Assuming it refers to $$ \sin 142^{\circ} $$:
   $$
   \sin 142^{\circ} = \sin(180^{\circ} - 38^{\circ}) = \sin 38^{\circ} = p
   $$

### Part (d)
Now, we express the given functions in terms of $$ k $$ where $$ \tan 200^{\circ} = k $$.

1. **For $$ \tan 20^{\circ} $$**:
   $$
   \tan 20^{\circ} = \frac{1}{\tan(200^{\circ} - 180^{\circ})} = \frac{1}{k}
   $$

2. **For $$ \tan 340^{\circ} $$**:
   $$
   \tan 340^{\circ} = \tan(360^{\circ} - 20^{\circ}) = -\tan 20^{\circ = -\frac{1}{k}}
   $$

3. **For $$ \tan(-160^{\circ}) $$**:
   $$
   \tan(-160^{\circ}) = -\tan(160^{\circ}) = -\tan(180^{\circ} - 20^{\circ}) = -(-\tan 20^{\circ}) = k
   $$

4. **For $$ \sin 200^{\circ} $$**:
   $$
   \sin 200^{\circ} = -\sin(20^{\circ}) = -\frac{k}{\sqrt{1 + k^2}}
   $$

5. **For $$ \cos 20^{\circ} $$**:
   $$
   \cos 20^{\circ} = \frac{1}{\sqrt{1 + k^2}}
   $$

### Part (b)
Now, we express the given functions in terms of $$ m $$ where $$ \cos 110^{\circ} = m $$.

1. **For $$ \cos 70^{\circ} $$**:
   $$
   \cos 70^{\circ} = \sin 20^{\circ} = \sqrt{1 - m^2}
   $$

2. **For $$ \cos 290^{\circ} $$**:
   $$
   \cos 290^{\circ} = \cos(360^{\circ} - 70^{\circ}) = \cos 70^{\circ} = \sqrt{1 - m^2}
   $$

3. **For $$ \cos 160^{\circ} $$**:
   $$
   \cos 160^{\circ} = -\cos(20^{\circ}) = -\sqrt{1 - m^2}
   $$

4. **For $$ \cos 20^{\circ} $$**:
   $$
   \cos 20^{\circ} = \sqrt{1 - m^2}
   $$

5. **For $$ 110^{+1} $$**:
   This seems to be a typo. Assuming it refers to $$ \cos(110^{\circ} + 1) $$, we cannot express it directly in terms of $$ m $$ without additional context.

### Summary
- Part (c) results in expressions involving $$ p $$.
- Part (d) results in expressions involving $$ k $$.
- Part (b) results in expressions involving $$ m $$.

If you need further clarification or specific calculations, please let me know!
Here are the simplified answers for each part: **Part (c):** 1. $$ \cos 322^{\circ} = p $$ 2. $$ t \ln 52^{\circ} $$ (no simplification possible) 3. $$ \sin 232^{\circ} = -\sin 52^{\circ} $$ 4. $$ t \tan 38^{\circ} = t \cdot \frac{p}{\sqrt{1 - p^2}} $$ 5. $$ \sha 142^{\circ} = p $$ **Part (d):** 1. $$ \tan 20^{\circ} = \frac{1}{k} $$ 2. $$ \tan 340^{\circ} = -\frac{1}{k} $$ 3. $$ \tan(-160^{\circ}) = k $$ 4. $$ \sin 200^{\circ} = -\frac{k}{\sqrt{1 + k^2}} $$ 5. $$ \cos 20^{\circ} = \frac{1}{\sqrt{1 + k^2}} $$ **Part (b):** 1. $$ \cos 70^{\circ} = \sqrt{1 - m^2} $$ 2. $$ \cos 290^{\circ} = \sqrt{1 - m^2} $$ 3. $$ \cos 160^{\circ} = -\sqrt{1 - m^2} $$ 4. $$ \cos 20^{\circ} = \sqrt{1 - m^2} $$ 5. $$ 110^{+1} $$ (no simplification possible) If you need further assistance or specific calculations, feel free to ask!

© If , write the following in terms of .
(1)
(2)
(J)
(4)
(5) sha
(d) If , write the following la terms of :
(I)
(2) .
(J)
(4)
(5) :
(b) If , write the following in terms of mi
(I)
(2)
(3)
(4).
(5)
)

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© If , write the following in terms of . (1) (2) (J) (4) (5) sha (d) If , write the following la terms of : (I) (2) . (J) (4) (5) : (b) If , write the following in terms of mi (I) (2) (3) (4). (5) )