To show that the sine or cosine value of an angle measuring \( 48^\circ \) is equivalent to the sine or cosine value of that same angle measured in radians, we first need to convert \( 48^\circ \) to radians.
The conversion from degrees to radians is given by the formula:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
For \( 48^\circ \):
\[ 48^\circ = 48 \times \frac{\pi}{180} = \frac{48\pi}{180} = \frac{48\pi}{180} = \frac{8\pi}{30} = \frac{4\pi}{15} \]
Now we have:
- \( 48^\circ \) in degrees is equivalent to \( \frac{4\pi}{15} \) in radians.
Next, we can express the sine and cosine values:
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Sine: \[ \sin(48^\circ) = \sin\left(\frac{4\pi}{15}\right) \]
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Cosine: \[ \cos(48^\circ) = \cos\left(\frac{4\pi}{15}\right) \]
At this stage, we do not have to calculate the numerical values of these sine and cosine functions. Both expressions \( \sin(48^\circ) \) and \( \sin\left(\frac{4\pi}{15}\right) \) represent the same value, and likewise for the cosine.
To provide the numerical value for either function, we can calculate:
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Using a calculator:
- \( \sin(48^\circ) \approx 0.7431 \)
- \( \cos(48^\circ) \approx 0.6691 \)
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Or, with radians:
- \( \sin\left(\frac{4\pi}{15}\right) \approx 0.7431 \)
- \( \cos\left(\frac{4\pi}{15}\right) \approx 0.6691 \)
In conclusion, we have shown that:
\[ \sin(48^\circ) = \sin\left(\frac{4\pi}{15}\right) \] and \[ \cos(48^\circ) = \cos\left(\frac{4\pi}{15}\right) \]
Both values are approximately:
\[ \sin(48^\circ) \approx 0.7431 \] \[ \cos(48^\circ) \approx 0.6691 \]