Question
Rewrite the sine function f(x)=−two sin(x−π)+three as a cosine function. WRITE WITH WORDS NOT NUMBERS explain how you arrived at your answer.
Answers
GPT-4o mini
To rewrite the sine function \( f(x) = -2 \sin(x - \pi) + 3 \) as a cosine function, we can use the relationship between sine and cosine. Specifically, we know that the sine function can be expressed in terms of cosine by using the identity that relates their values:
The sine of an angle can be represented as the cosine of the complementary angle: \( \sin(x) = \cos\left(\frac{\pi}{2} - x\right) \). This means that for any angle \( a \), we can write \( \sin(a) = \cos\left(\frac{\pi}{2} - a\right) \).
Now, let's break down the transformation of the original function step by step:
1. **Start with the Original Function:** We have \( f(x) = -2 \sin(x - \pi) + 3 \).
2. **Transform the Sine to Cosine:** To change the sine term into a cosine term, we can focus on the argument of the sine function, which is \( x - \pi \). We can rewrite \( \sin(x - \pi) \) as \( -\cos(x - \pi - \frac{\pi}{2}) \). This follows because the sine of an angle is equivalent to the negative cosine of that angle plus a quarter turn (or to the left on the unit circle).
3. **Apply the Transformation:** By substituting, we get:
\[
\sin(x - \pi) = -\cos\left(x - \pi + \frac{\pi}{2}\right)
\]
When simplified, this becomes:
\[
\sin(x - \pi) = -\cos\left(x - \frac{\pi}{2}\right)
\]
4. **Substitute Back into the Function:** Now, substituting this expression back into our original function:
\[
f(x) = -2\left(-\cos\left(x - \frac{\pi}{2}\right)\right) + 3
\]
This simplifies to:
\[
f(x) = 2 \cos\left(x - \frac{\pi}{2}\right) + 3
\]
5. **Final Result:** Therefore, the sine function can be rewritten in terms of cosine as:
\[
f(x) = 2 \cos\left(x - \frac{\pi}{2}\right) + 3
\]
In summary, we transformed the sine function into a cosine function by using the fundamental relationship between the two functions and adjusting the argument accordingly. The final expression shows how the amplitude and the vertical shift of the original sine function translate to the new cosine formulation.
The sine of an angle can be represented as the cosine of the complementary angle: \( \sin(x) = \cos\left(\frac{\pi}{2} - x\right) \). This means that for any angle \( a \), we can write \( \sin(a) = \cos\left(\frac{\pi}{2} - a\right) \).
Now, let's break down the transformation of the original function step by step:
1. **Start with the Original Function:** We have \( f(x) = -2 \sin(x - \pi) + 3 \).
2. **Transform the Sine to Cosine:** To change the sine term into a cosine term, we can focus on the argument of the sine function, which is \( x - \pi \). We can rewrite \( \sin(x - \pi) \) as \( -\cos(x - \pi - \frac{\pi}{2}) \). This follows because the sine of an angle is equivalent to the negative cosine of that angle plus a quarter turn (or to the left on the unit circle).
3. **Apply the Transformation:** By substituting, we get:
\[
\sin(x - \pi) = -\cos\left(x - \pi + \frac{\pi}{2}\right)
\]
When simplified, this becomes:
\[
\sin(x - \pi) = -\cos\left(x - \frac{\pi}{2}\right)
\]
4. **Substitute Back into the Function:** Now, substituting this expression back into our original function:
\[
f(x) = -2\left(-\cos\left(x - \frac{\pi}{2}\right)\right) + 3
\]
This simplifies to:
\[
f(x) = 2 \cos\left(x - \frac{\pi}{2}\right) + 3
\]
5. **Final Result:** Therefore, the sine function can be rewritten in terms of cosine as:
\[
f(x) = 2 \cos\left(x - \frac{\pi}{2}\right) + 3
\]
In summary, we transformed the sine function into a cosine function by using the fundamental relationship between the two functions and adjusting the argument accordingly. The final expression shows how the amplitude and the vertical shift of the original sine function translate to the new cosine formulation.