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 \( 48^\circ \) 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:

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

2. **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:

- Using a calculator:
- \( \sin(48^\circ) \approx 0.7431 \)
- \( \cos(48^\circ) \approx 0.6691 \)

- 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:

\[
\sin(48^\circ) \approx 0.7431
\]
\[
\cos(48^\circ) \approx 0.6691
\]