To show that the sine and cosine values of an angle measuring 48 degrees are equivalent to the sine and cosine values of the same angle measured in radians, we first need to convert 48 degrees to radians.
The conversion from degrees to radians is done using the formula:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Thus, for 48 degrees:
\[ 48 \text{ degrees} = 48 \times \frac{\pi}{180} = \frac{48\pi}{180} = \frac{4\pi}{15} \text{ radians} \]
Now, we can evaluate the sine and cosine for both the degree and radian measures.
- Sine value:
Using 48 degrees: \[ \sin(48^\circ) \]
Using \(\frac{4\pi}{15}\) radians: \[ \sin\left(\frac{4\pi}{15}\right) \]
Both of these expressions represent the sine of the same angle measured in different units. Thus, we have:
\[ \sin(48^\circ) = \sin\left(\frac{4\pi}{15}\right) \]
- Cosine value:
Using 48 degrees: \[ \cos(48^\circ) \]
Using \(\frac{4\pi}{15}\) radians: \[ \cos\left(\frac{4\pi}{15}\right) \]
Similarly, we can state: \[ \cos(48^\circ) = \cos\left(\frac{4\pi}{15}\right) \]
Since sine and cosine functions are periodic functions with the same value for an angle expressed in degrees and radians, we establish that:
\[ \sin(48^\circ) = \sin\left(\frac{4\pi}{15}\right) \] \[ \cos(48^\circ) = \cos\left(\frac{4\pi}{15}\right) \]
Evaluating the Values
If you need the actual numerical values of sine and cosine for 48 degrees (or \(\frac{4\pi}{15}\)), we can use a calculator or trigonometric tables:
\[ \sin(48^\circ) \approx 0.7431 \] \[ \cos(48^\circ) \approx 0.6691 \]
And thus, the final values are:
\[ \sin(48^\circ) \approx \sin\left(\frac{4\pi}{15}\right) \approx 0.7431 \] \[ \cos(48^\circ) \approx \cos\left(\frac{4\pi}{15}\right) \approx 0.6691 \]
In conclusion, we have demonstrated that the sine and cosine values remain consistent under both measures (degrees and radians), and their approximate values are \(0.7431\) and \(0.6691\) respectively.
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