Question
Which of the following best approximates $\sin(0.5^\circ)$?
The correct answer is
$0.5 \times \frac{\pi}{180}$
Approximating sin(0.5 degrees)
For small angles $\theta$ measured in radians, the approximation $\sin(\theta) \approx \theta$ holds true.
Angle Conversion
The given angle is $0.5^\circ$. We need to convert this angle from degrees to radians.
The formula for conversion is:
Radians = Degrees $\times \frac{\pi}{180}$
Therefore, $0.5^\circ$ in radians is:
$0.5 \times \frac{\pi}{180}$
Applying Approximation
Using the small angle approximation $\sin(\theta) \approx \theta$ with $\theta$ in radians:
$\sin(0.5^\circ) \approx 0.5 \times \frac{\pi}{180}$
Matching with Options
Comparing the result $0.5 \times \frac{\pi}{180}$ with the provided options:
- Option 1: $0.5$
- Option 2: $0.5 \times \frac{\pi}{90}$
- Option 3: $0.5 \times \frac{\pi}{180}$
- Option 4: $0.5 \times \frac{\pi}{360}$
The approximation $0.5 \times \frac{\pi}{180}$ matches Option 3.
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