Question
WRITE IN WORDS NOT NUMBERS
Rewrite the sine function f(x)=−2sin(x−π)+3 as a cosine function. explain how you arrived at your answer.
Rewrite the sine function f(x)=−2sin(x−π)+3 as a cosine function. explain how you arrived at your answer.
Answers
Answered by
GPT-4o mini
To rewrite the sine function \( f(x) = -2\sin(x - \pi) + 3 \) as a cosine function, we can utilize the relationship between sine and cosine. Specifically, we can use the fact that the sine function can be expressed in terms of the cosine function using the identity:
\[
\sin(x) = \cos\left(x - \frac{\pi}{2}\right)
\]
We also need to consider the sign and the phase shift in the original sine function.
1. **Identify the phase shift**: The sine function in \( f(x) \) has a phase shift of \( x - \pi \). To express this using cosine, we need to adjust accordingly.
2. **Adjust the sine function**: Since we have a negative sign in front of the sine function, we can rewrite the sine part:
\[
-\sin(x - \pi) = \sin\left(-(x - \pi)\right) = -\sin(x - \pi)
\]
Using the sine-cosine relationship, we know that:
\[
-\sin(y) = \cos\left(y + \frac{\pi}{2}\right)
\]
So, for \( y = x - \pi \), we have:
\[
-\sin(x - \pi) = \cos\left((x - \pi) + \frac{\pi}{2}\right) = \cos\left(x - \frac{\pi}{2}\right)
\]
3. **Substituting back into the function**: Now we substitute this back into the original function:
\[
f(x) = -2\sin(x - \pi) + 3 = -2\left(-\sin(x - \pi)\right) + 3 = 2\cos\left(x - \frac{\pi}{2}\right) + 3
\]
4. **Final rewriting**: Therefore, we can rewrite the entire function as:
\[
f(x) = 2\cos\left(x - \frac{\pi}{2}\right) + 3
\]
In conclusion, \( f(x) = -2\sin(x - \pi) + 3 \) can be rewritten as \( f(x) = 2\cos\left(x - \frac{\pi}{2}\right) + 3 \) by utilizing the sine to cosine transformation and adjusting for phase shifts and the negative sign.
\[
\sin(x) = \cos\left(x - \frac{\pi}{2}\right)
\]
We also need to consider the sign and the phase shift in the original sine function.
1. **Identify the phase shift**: The sine function in \( f(x) \) has a phase shift of \( x - \pi \). To express this using cosine, we need to adjust accordingly.
2. **Adjust the sine function**: Since we have a negative sign in front of the sine function, we can rewrite the sine part:
\[
-\sin(x - \pi) = \sin\left(-(x - \pi)\right) = -\sin(x - \pi)
\]
Using the sine-cosine relationship, we know that:
\[
-\sin(y) = \cos\left(y + \frac{\pi}{2}\right)
\]
So, for \( y = x - \pi \), we have:
\[
-\sin(x - \pi) = \cos\left((x - \pi) + \frac{\pi}{2}\right) = \cos\left(x - \frac{\pi}{2}\right)
\]
3. **Substituting back into the function**: Now we substitute this back into the original function:
\[
f(x) = -2\sin(x - \pi) + 3 = -2\left(-\sin(x - \pi)\right) + 3 = 2\cos\left(x - \frac{\pi}{2}\right) + 3
\]
4. **Final rewriting**: Therefore, we can rewrite the entire function as:
\[
f(x) = 2\cos\left(x - \frac{\pi}{2}\right) + 3
\]
In conclusion, \( f(x) = -2\sin(x - \pi) + 3 \) can be rewritten as \( f(x) = 2\cos\left(x - \frac{\pi}{2}\right) + 3 \) by utilizing the sine to cosine transformation and adjusting for phase shifts and the negative sign.