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 angles in radians, sin(θ) ≈ θ and tan(θ) ≈ θ when θ is small (typically less than about 15 degrees).

### Step 2: Convert Degrees to Radians
We need to convert the angles from degrees to radians because the small angle approximation is valid in radians.

**(A) For sin 8 degrees:**
1. Convert 8 degrees to radians:
\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} = 8 \times \frac{\pi}{180} = \frac{8\pi}{180} = \frac{2\pi}{45} \]

**(B) For tan 5 degrees:**
2. Convert 5 degrees to radians:
\[ \text{Radians} = 5 \times \frac{\pi}{180} = \frac{5\pi}{180} = \frac{\pi}{36} \]

### Step 3: Apply the Small Angle Approximation
Using the small angle approximation, we can now find the approximate values.

**(A) For sin 8 degrees:**
3. Using the approximation:
\[ \sin(8^\circ) \approx \frac{2\pi}{45} \approx 0.139 \]
Rounding this value gives:
\[ \sin(8^\circ) \approx 0.14 \]

**(B) For tan 5 degrees:**
4. Using the approximation:
\[ \tan(5^\circ) \approx \frac{\pi}{36} \approx 0.087 \]
Rounding this value gives:
\[ \tan(5^\circ) \approx 0.09 \]

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