1. If $$ \sin 40^{\circ}=p $$ write the following in terms of $$ p $$. $$ \begin{array}{llll} ext { (a) } \sin 140^{\circ} & ext { (b) } \sin 220^{\circ} & ext { (c) } \sin \left(-140^{\circ} ight) \ ext { (d) } \cos 50^{\circ} & ext { (c) } \sin 50^{\circ} & ext { (f) } an 220^{\circ}\end{array} $$ If $$ \cos 35^{\circ}=k $$ write the following in terms of $$ k $$. $$ \begin{array}{lll} ext { (a) } \cos 215^{\circ} & ext { (b) } \sin 55^{\circ} & ext { (c) } \cos \left(-35^{\circ} ight) \ ext { (d) } \sin 665^{\circ} & ext { (c) } \sin 325^{\circ} & \end{array} $$ If $$ an 22^{\circ}=t $$ write the following in terms of $$ t $$. $$ \begin{array}{lll} ext { (a) } an 202^{\circ} & ext { (b) } an 338^{\circ} & ext { (c) } an \left(-22^{\circ} ight) \ ext { (d) } an 518^{\circ} & ext { (e) } \cos 22^{\circ} & \end{array} $$

Question

1. If $$ \sin 40^{\circ}=p $$ write the following in terms of $$ p $$. $$ \begin{array}{llll}	ext { (a) } \sin 140^{\circ} & 	ext { (b) } \sin 220^{\circ} & 	ext { (c) } \sin \left(-140^{\circ}ight) \ 	ext { (d) } \cos 50^{\circ} & 	ext { (c) } \sin 50^{\circ} & 	ext { (f) } 	an 220^{\circ}\end{array} $$ If $$ \cos 35^{\circ}=k $$ write the following in terms of $$ k $$. $$ \begin{array}{lll}	ext { (a) } \cos 215^{\circ} & 	ext { (b) } \sin 55^{\circ} & 	ext { (c) } \cos \left(-35^{\circ}ight) \ 	ext { (d) } \sin 665^{\circ} & 	ext { (c) } \sin 325^{\circ} & \end{array} $$ If $$ 	an 22^{\circ}=t $$ write the following in terms of $$ t $$. $$ \begin{array}{lll}	ext { (a) } 	an 202^{\circ} & 	ext { (b) } 	an 338^{\circ} & 	ext { (c) } 	an \left(-22^{\circ}ight) \ 	ext { (d) } 	an 518^{\circ} & 	ext { (e) } \cos 22^{\circ} & \end{array} $$

UpStudy AI · Accepted Answer

Let's solve each part step by step, using the known values of $$ p $$, $$ k $$, and $$ t $$.

### For $$ \sin 40^{\circ} = p $$:

1. **(a) $$ \sin 140^{\circ} $$**:
   $$
   \sin(140^{\circ}) = \sin(180^{\circ} - 40^{\circ}) = \sin(40^{\circ}) = p
   $$

2. **(b) $$ \sin 220^{\circ} $$**:
   $$
   \sin(220^{\circ}) = \sin(180^{\circ} + 40^{\circ}) = -\sin(40^{\circ}) = -p
   $$

3. **(c) $$ \sin(-140^{\circ}) $$**:
   $$
   \sin(-140^{\circ}) = -\sin(140^{\circ}) = -p
   $$

4. **(d) $$ \cos 50^{\circ} $$**:
   $$
   \cos(50^{\circ}) = \sin(90^{\circ} - 50^{\circ}) = \sin(40^{\circ}) = p
   $$

5. **(e) $$ \sin 50^{\circ} $$**:
   $$
   \sin(50^{\circ}) = \sin(90^{\circ} - 40^{\circ}) = \cos(40^{\circ}) = \sqrt{1 - p^2}
   $$

6. **(f) $$ \tan 220^{\circ} $$**:
   $$
   \tan(220^{\circ}) = \frac{\sin(220^{\circ})}{\cos(220^{\circ})} = \frac{-p}{-\cos(40^{\circ})} = \frac{p}{\sqrt{1 - p^2}}
   $$

### For $$ \cos 35^{\circ} = k $$:

1. **(a) $$ \cos 215^{\circ} $$**:
   $$
   \cos(215^{\circ}) = \cos(180^{\circ} + 35^{\circ}) = -\cos(35^{\circ}) = -k
   $$

2. **(b) $$ \sin 55^{\circ} $$**:
   $$
   \sin(55^{\circ}) = \sin(90^{\circ} - 35^{\circ}) = \cos(35^{\circ}) = k
   $$

3. **(c) $$ \cos(-35^{\circ}) $$**:
   $$
   \cos(-35^{\circ}) = \cos(35^{\circ}) = k
   $$

4. **(d) $$ \sin 665^{\circ} $$**:
   $$
   \sin(665^{\circ}) = \sin(665^{\circ} - 360^{\circ}) = \sin(305^{\circ}) = -\sin(25^{\circ}) = -\cos(65^{\circ}) = -\sin(90^{\circ} - 25^{\circ}) = -\cos(25^{\circ})
   $$

5. **(e) $$ \sin 325^{\circ} $$**:
   $$
   \sin(325^{\circ}) = \sin(360^{\circ} - 35^{\circ}) = -\sin(35^{\circ}) = -\sqrt{1 - k^2}
   $$

### For $$ \tan 22^{\circ} = t $$:

1. **(a) $$ \tan 202^{\circ} $$**:
   $$
   \tan(202^{\circ}) = \tan(180^{\circ} + 22^{\circ}) = \tan(22^{\circ}) = t
   $$

2. **(b) $$ \tan 338^{\circ} $$**:
   $$
   \tan(338^{\circ}) = \tan(360^{\circ} - 22^{\circ}) = -\tan(22^{\circ}) = -t
   $$

3. **(c) $$ \tan(-22^{\circ}) $$**:
   $$
   \tan(-22^{\circ}) = -\tan(22^{\circ}) = -t
   $$

4. **(d) $$ \tan 518^{\circ} $$**:
   $$
   \tan(518^{\circ}) = \tan(518^{\circ} - 360^{\circ}) = \tan(158^{\circ}) = -\tan(22^{\circ}) = -t
   $$

5. **(e) $$ \cos 22^{\circ} $$**:
   $$
   \cos(22^{\circ}) = \frac{1}{\sqrt{1 + \tan^2(22^{\circ})}} = \frac{1}{\sqrt{1 + t^2}}
   $$

### Summary of Results:
- For $$ \sin 40^{\circ} = p $$:
  - (a) $$ p $$
  - (b) $$ -p $$
  - (c) $$ -p $$
  - (d) $$ p $$
  - (e) $$ \sqrt{1 - p^2} $$
  - (f) $$ \frac{p}{\sqrt{1 - p^2}} $$

- For $$ \cos 35^{\circ} = k $$:
  - (a) $$ -k $$
  - (b) $$ k $$
  - (c) $$ k $$
  - (d) $$ -\cos(25^{\circ}) $$
  - (e) $$ -\sqrt{1 - k^2} $$

- For $$ \tan 22^{\circ} = t $$:
  - (a) $$ t $$
  - (b) $$ -t $$
  - (c) $$ -t $$
  - (d) $$ -t $$
  - (e) $$ \frac{1}{\sqrt{1 + t^2}} $$
Here are the expressions in terms of $$ p $$, $$ k $$, and $$ t $$: - **For $$ \sin 40^{\circ} = p $$:** - (a) $$ p $$ - (b) $$ -p $$ - (c) $$ -p $$ - (d) $$ p $$ - (e) $$ \sqrt{1 - p^2} $$ - (f) $$ \frac{p}{\sqrt{1 - p^2}} $$ - **For $$ \cos 35^{\circ} = k $$:** - (a) $$ -k $$ - (b) $$ k $$ - (c) $$ k $$ - (d) $$ -\cos(25^{\circ}) $$ - (e) $$ -\sqrt{1 - k^2} $$ - **For $$ \tan 22^{\circ} = t $$:** - (a) $$ t $$ - (b) $$ -t $$ - (c) $$ -t $$ - (d) $$ -t $$ - (e) $$ \frac{1}{\sqrt{1 + t^2}} $$

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