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.(2 points)
Short Answer Rubric (2 points)
Points Awarded Criteria
2
The answer is very clear.
The answer is fully supported by details.
There may be a few errors in grammar, usage, or mechanics. Errors do not interfere with meaning.
1
The answer is somewhat clear.
Details are weak or not related.
There are some errors in grammar, usage, and mechanics. Errors interfere with meaning.
0 The question is not answered.
is equivalent to the sine or cosine value of that same angle measured in radians and state the value of the function.(2 points)
Short Answer Rubric (2 points)
Points Awarded Criteria
2
The answer is very clear.
The answer is fully supported by details.
There may be a few errors in grammar, usage, or mechanics. Errors do not interfere with meaning.
1
The answer is somewhat clear.
Details are weak or not related.
There are some errors in grammar, usage, and mechanics. Errors interfere with meaning.
0 The question is not answered.
Answers
GPT-4o mini
To demonstrate 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, we should first convert 48° to radians. The conversion formula between degrees and radians is:
\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\]
For 48°:
\[
48° \times \frac{\pi}{180} = \frac{48\pi}{180} = \frac{24\pi}{90} = \frac{8\pi}{30} = \frac{4\pi}{15}
\]
Now we have the angle 48° expressed in radians as \(\frac{4\pi}{15}\).
Next, we compare the sine and cosine values of both representations of the angle:
\[
\sin(48°) = \sin\left(\frac{4\pi}{15}\right)
\]
\[
\cos(48°) = \cos\left(\frac{4\pi}{15}\right)
\]
Since trigonometric functions are periodic and defined for both degree and radian measures, we can conclude that:
\[
\sin(48°) = \sin\left(\frac{4\pi}{15}\right) \quad \text{and} \quad \cos(48°) = \cos\left(\frac{4\pi}{15}\right)
\]
To state the values of these functions, we can use a calculator.
Calculating the sine and cosine values:
\[
\sin(48°) \approx 0.7431 \quad \text{and} \quad \cos(48°) \approx 0.6691
\]
Thus, the sine and cosine values are:
- \(\sin(48°) \approx 0.7431\)
- \(\cos(48°) \approx 0.6691\)
In conclusion, the sine and cosine values of 48° are indeed equivalent to the sine and cosine of the same angle expressed in radians (\(\frac{4\pi}{15}\)), confirming the relationship.
\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\]
For 48°:
\[
48° \times \frac{\pi}{180} = \frac{48\pi}{180} = \frac{24\pi}{90} = \frac{8\pi}{30} = \frac{4\pi}{15}
\]
Now we have the angle 48° expressed in radians as \(\frac{4\pi}{15}\).
Next, we compare the sine and cosine values of both representations of the angle:
\[
\sin(48°) = \sin\left(\frac{4\pi}{15}\right)
\]
\[
\cos(48°) = \cos\left(\frac{4\pi}{15}\right)
\]
Since trigonometric functions are periodic and defined for both degree and radian measures, we can conclude that:
\[
\sin(48°) = \sin\left(\frac{4\pi}{15}\right) \quad \text{and} \quad \cos(48°) = \cos\left(\frac{4\pi}{15}\right)
\]
To state the values of these functions, we can use a calculator.
Calculating the sine and cosine values:
\[
\sin(48°) \approx 0.7431 \quad \text{and} \quad \cos(48°) \approx 0.6691
\]
Thus, the sine and cosine values are:
- \(\sin(48°) \approx 0.7431\)
- \(\cos(48°) \approx 0.6691\)
In conclusion, the sine and cosine values of 48° are indeed equivalent to the sine and cosine of the same angle expressed in radians (\(\frac{4\pi}{15}\)), confirming the relationship.