(b) If $$ \sin 10^{\circ}=m $$, write the following in terms of $$ m $$ : $$ \begin{array}{lll} ext { (1) } \sin 350^{\circ} & ext { (2) } \sin \left(-10^{\circ} ight) & ext { (3) } \cos 260^{\circ} \ ext { (4) } \cos 10^{\circ} & ext { (5) } an 170^{\circ} & \ ext { (c) If } \cos 38^{\circ}=p ext {, write the following in terms of } p ext { : } \ ext { (1) } \cos 322^{\circ} & ext { (2) } \sin 52^{\circ} & ext { (3) } \sin 232^{\circ} \ ext { (4) } an 38^{\circ} & ext { (5) } \sin 142^{\circ} & \end{array} $$

Question

(b) If $$ \sin 10^{\circ}=m $$, write the following in terms of $$ m $$ : $$ \begin{array}{lll}	ext { (1) } \sin 350^{\circ} & 	ext { (2) } \sin \left(-10^{\circ}ight) & 	ext { (3) } \cos 260^{\circ} \ 	ext { (4) } \cos 10^{\circ} & 	ext { (5) } 	an 170^{\circ} & \ 	ext { (c) If } \cos 38^{\circ}=p 	ext {, write the following in terms of } p 	ext { : } \ 	ext { (1) } \cos 322^{\circ} & 	ext { (2) } \sin 52^{\circ} & 	ext { (3) } \sin 232^{\circ} \ 	ext { (4) } 	an 38^{\circ} & 	ext { (5) } \sin 142^{\circ} & \end{array} $$

UpStudy AI · Accepted Answer

To express the given trigonometric functions in terms of $$ m = \sin 10^{\circ} $$ and $$ p = \cos 38^{\circ} $$, we will use trigonometric identities and properties.

### Part (b)

1. **For $$ \sin 350^{\circ} $$**:
   $$
   \sin 350^{\circ} = \sin(360^{\circ} - 10^{\circ}) = -\sin 10^{\circ} = -m
   $$

2. **For $$ \sin(-10^{\circ}) $$**:
   $$
   \sin(-10^{\circ}) = -\sin 10^{\circ} = -m
   $$

3. **For $$ \cos 260^{\circ} $$**:
   $$
   \cos 260^{\circ} = \cos(270^{\circ} - 10^{\circ}) = -\sin 10^{\circ} = -m
   $$

4. **For $$ \cos 10^{\circ} $$**:
   Using the identity $$ \cos^2 \theta + \sin^2 \theta = 1 $$:
   $$
   \cos 10^{\circ} = \sqrt{1 - \sin^2 10^{\circ}} = \sqrt{1 - m^2}
   $$

5. **For $$ \tan 170^{\circ} $$**:
   $$
   \tan 170^{\circ} = \tan(180^{\circ} - 10^{\circ}) = -\tan 10^{\circ} = -\frac{\sin 10^{\circ}}{\cos 10^{\circ}} = -\frac{m}{\sqrt{1 - m^2}}
   $$

### Part (c)

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

2. **For $$ \sin 52^{\circ} $$**:
   $$
   \sin 52^{\circ} = \sin(90^{\circ} - 38^{\circ}) = \cos 38^{\circ} = p
   $$

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

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

5. **For $$ \sin 142^{\circ} $$**:
   $$
   \sin 142^{\circ} = \sin(180^{\circ} - 38^{\circ}) = \sin 38^{\circ} = \sqrt{1 - p^2}
   $$

### Summary of Results

- Part (b):
  1. $$ \sin 350^{\circ} = -m $$
  2. $$ \sin(-10^{\circ}) = -m $$
  3. $$ \cos 260^{\circ} = -m $$
  4. $$ \cos 10^{\circ} = \sqrt{1 - m^2} $$
  5. $$ \tan 170^{\circ} = -\frac{m}{\sqrt{1 - m^2}} $$

- Part (c):
  1. $$ \cos 322^{\circ} = p $$
  2. $$ \sin 52^{\circ} = p $$
  3. $$ \sin 232^{\circ} = -p $$
  4. $$ \tan 38^{\circ} = \frac{\sqrt{1 - p^2}}{p} $$
  5. $$ \sin 142^{\circ} = \sqrt{1 - p^2} $$
- **Part (b):** 1. $$ \sin 350^{\circ} = -m $$ 2. $$ \sin(-10^{\circ}) = -m $$ 3. $$ \cos 260^{\circ} = -m $$ 4. $$ \cos 10^{\circ} = \sqrt{1 - m^2} $$ 5. $$ \tan 170^{\circ} = -\frac{m}{\sqrt{1 - m^2}} $$ - **Part (c):** 1. $$ \cos 322^{\circ} = p $$ 2. $$ \sin 52^{\circ} = p $$ 3. $$ \sin 232^{\circ} = -p $$ 4. $$ \tan 38^{\circ} = \frac{\sqrt{1 - p^2}}{p} $$ 5. $$ \sin 142^{\circ} = \sqrt{1 - p^2} $$

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