Question

Show that the sine or cosine value of an angle measuring 48° is equivalent to the sine or cosine value of that same angle measured in radians and state the value of the function.

Answer 1

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:

1. **Sine Function**:
\[
\sin(48^\circ) = \sin\left(\frac{4\pi}{15}\right)
\]

2. **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:
- \( \sin(48^\circ) \approx 0.7431 \)
- \( \cos(48^\circ) \approx 0.6691 \)