Use the small angle appoximate to find approximate values for (A) `sin 8^(@) ` and (B) `tan 5^(@)`

Answer

To solve the problem using the small angle approximation, we will follow these steps:

### Step 1: Understand the Small Angle Approximation
The small angle approximation states that for small angles (in radians), the sine and tangent of the angle can be approximated as:
- \( \sin \theta \approx \theta \)
- \( \tan \theta \approx \theta \)

### Step 2: Convert Degrees to Radians
To use the small angle approximation, we need to convert the angles from degrees to radians.

#### (A) For \( \sin 8^\circ \):
1. Convert \( 8^\circ \) to radians using the conversion factor \( \frac{\pi}{180} \):
\[ 8^\circ = 8 \times \frac{\pi}{180} = \frac{8\pi}{180} = \frac{2\pi}{45} \text{ radians} \]

#### (B) For \( \tan 5^\circ \):
2. Convert \( 5^\circ \) to radians using the same conversion factor:
\[ 5^\circ = 5 \times \frac{\pi}{180} = \frac{5\pi}{180} = \frac{\pi}{36} \text{ radians} \]

### Step 3: Apply the Small Angle Approximation
Now we can apply the small angle approximation to find the approximate values.

#### (A) For \( \sin 8^\circ \):
3. Using the approximation:
\[ \sin 8^\circ \approx \frac{2\pi}{45} \]
Now, we can calculate \( \frac{2\pi}{45} \):
\[ \frac{2\pi}{45} \approx \frac{2 \times 3.14}{45} \approx \frac{6.28}{45} \approx 0.139 \]
Thus, \( \sin 8^\circ \approx 0.14 \).

#### (B) For \( \tan 5^\circ \):
4. Using the approximation:
\[ \tan 5^\circ \approx \frac{\pi}{36} \]
Now, we can calculate \( \frac{\pi}{36} \):
\[ \frac{\pi}{36} \approx \frac{3.14}{36} \approx 0.087 \]
Thus, \( \tan 5^\circ \approx 0.09 \).

### Final Answers:
- \( \sin 8^\circ \approx 0.14 \)
- \( \tan 5^\circ \approx 0.09 \)