To show that the sine or cosine value of an angle measuring 48° is equivalent to the sine or cosine value of that same angle in radians, we first need to convert 48 degrees to radians.
The conversion formula from degrees to radians is:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
For 48 degrees, we have:
\[ 48^\circ \times \frac{\pi}{180} = \frac{48\pi}{180} \]
This simplifies to:
\[ \frac{48\pi}{180} = \frac{8\pi}{30} = \frac{4\pi}{15} \]
Now we can confirm that the sine and cosine functions give the same numerical values in both degrees and radians:
-
Sine Function: \[ \sin(48^\circ) = \sin\left(\frac{4\pi}{15}\right) \]
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Cosine Function: \[ \cos(48^\circ) = \cos\left(\frac{4\pi}{15}\right) \]
Now, we can calculate the sine and cosine values.
Using a calculator (or trigonometric tables):
- \(\sin(48^\circ) \approx 0.7431\)
- \(\cos(48^\circ) \approx 0.6691\)
Since \(\sin\) and \(\cos\) functions only depend on the angle, the values calculated for \(\sin\left(\frac{4\pi}{15}\right)\) and \(\cos\left(\frac{4\pi}{15}\right)\) will be the same as those from \(48^\circ\).
Thus:
- \(\sin\left(\frac{4\pi}{15}\right) \approx 0.7431\)
- \(\cos\left(\frac{4\pi}{15}\right) \approx 0.6691\)
Therefore, we have shown that \( \sin(48^\circ) = \sin\left(\frac{4\pi}{15}\right) \) and \( \cos(48^\circ) = \cos\left(\frac{4\pi}{15}\right) \) with the numerical values approximately equal to:
- \( \sin(48^\circ) \approx 0.7431 \)
- \( \cos(48^\circ) \approx 0.6691 \)
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