To demonstrate that the sine or cosine value of an angle measured in degrees is equivalent to the value of the same angle measured in radians, we first need to convert the angle from degrees to radians.
The conversion from degrees to radians is done using the formula:
\[ \text{radians} = \text{degrees} \times \left(\frac{\pi}{180}\right) \]
For an angle of \( 48 \) degrees:
\[ 48 \text{ degrees} = 48 \times \left(\frac{\pi}{180}\right) = \frac{48\pi}{180} = \frac{8\pi}{30} = \frac{4\pi}{15} \text{ radians} \]
Next, we can evaluate \( \sin(48^\circ) \) and \( \sin\left(\frac{4\pi}{15}\right) \):
\[ \sin(48^\circ) \quad \text{and} \quad \sin\left(\frac{4\pi}{15}\right) \]
Since \( \sin(48^\circ) \) is equal to the sine of the angle when converted to radians, we have:
\[ \sin(48^\circ) = \sin\left(\frac{4\pi}{15}\right) \]
The same applies to the cosine function:
\[ \cos(48^\circ) = \cos\left(\frac{4\pi}{15}\right) \]
To find the numerical values, we can compute:
\[ \sin(48^\circ) \approx 0.7431 \quad \text{and} \quad \sin\left(\frac{4\pi}{15}\right) \approx 0.7431 \]
Thus, we confirm that:
\[ \sin(48^\circ) = \sin\left(\frac{4\pi}{15}\right) \text{ and } \cos(48^\circ) = \cos\left(\frac{4\pi}{15}\right) \]
In conclusion, the sine of \( 48 \) degrees is equivalent to the sine of \( \frac{4\pi}{15} \) radians, and both values approximately equal \( 0.7431 \). The exact function values can be derived using a scientific calculator or sine function tables.
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