Question
Which trigonometric function is equivalent to lower f left parenthesis x right parenthesis equals sine x?
(1 point)
Responses
lower f left parenthesis x right parenthesis equals cosine left parenthesis x minus start fraction 3 pi over 2 end fraction right parenthesis
Image with alt text: lower f left parenthesis x right parenthesis equals cosine left parenthesis x minus start fraction 3 pi over 2 end fraction right parenthesis
lower f left parenthesis x right parenthesis equals cosine left parenthesis x minus start fraction pi over 2 end fraction right parenthesis
Image with alt text: lower f left parenthesis x right parenthesis equals cosine left parenthesis x minus start fraction pi over 2 end fraction right parenthesis
lower f left parenthesis x right parenthesis equals cosine left parenthesis negative x minus start fraction pi over 2 end fraction right parenthesis
Image with alt text: lower f left parenthesis x right parenthesis equals cosine left parenthesis negative x minus start fraction pi over 2 end fraction right parenthesis
lower f left parenthesis x right parenthesis equals cosine left parenthesis x plus pi right parenthesis
(1 point)
Responses
lower f left parenthesis x right parenthesis equals cosine left parenthesis x minus start fraction 3 pi over 2 end fraction right parenthesis
Image with alt text: lower f left parenthesis x right parenthesis equals cosine left parenthesis x minus start fraction 3 pi over 2 end fraction right parenthesis
lower f left parenthesis x right parenthesis equals cosine left parenthesis x minus start fraction pi over 2 end fraction right parenthesis
Image with alt text: lower f left parenthesis x right parenthesis equals cosine left parenthesis x minus start fraction pi over 2 end fraction right parenthesis
lower f left parenthesis x right parenthesis equals cosine left parenthesis negative x minus start fraction pi over 2 end fraction right parenthesis
Image with alt text: lower f left parenthesis x right parenthesis equals cosine left parenthesis negative x minus start fraction pi over 2 end fraction right parenthesis
lower f left parenthesis x right parenthesis equals cosine left parenthesis x plus pi right parenthesis
Answers
GPT-4o mini
To determine which function is equivalent to \( f(x) = \sin x \), we can use the properties of sine and cosine functions, particularly their phase shifts.
The sine function can be expressed in terms of the cosine function with a phase shift. Specifically:
\[
\sin x = \cos\left(x - \frac{\pi}{2}\right)
\]
This means that sine can be rewritten as the cosine of an angle shifted by \(-\frac{\pi}{2}\).
Now let's analyze the responses:
1. \( f(x) = \cos\left(x - \frac{3\pi}{2}\right) \)
- This is not equivalent to \( \sin x \).
2. \( f(x) = \cos\left(x - \frac{\pi}{2}\right) \)
- This is equivalent to \( \sin x \).
3. \( f(x) = \cos\left(-x - \frac{\pi}{2}\right) \)
- This is not equivalent to \( \sin x \).
4. \( f(x) = \cos(x + \pi) \)
- This is equal to \(-\cos x\), which is also not equivalent to \( \sin x \).
The correct answer is:
**\( f(x) = \cos\left(x - \frac{\pi}{2}\right) \)**
The sine function can be expressed in terms of the cosine function with a phase shift. Specifically:
\[
\sin x = \cos\left(x - \frac{\pi}{2}\right)
\]
This means that sine can be rewritten as the cosine of an angle shifted by \(-\frac{\pi}{2}\).
Now let's analyze the responses:
1. \( f(x) = \cos\left(x - \frac{3\pi}{2}\right) \)
- This is not equivalent to \( \sin x \).
2. \( f(x) = \cos\left(x - \frac{\pi}{2}\right) \)
- This is equivalent to \( \sin x \).
3. \( f(x) = \cos\left(-x - \frac{\pi}{2}\right) \)
- This is not equivalent to \( \sin x \).
4. \( f(x) = \cos(x + \pi) \)
- This is equal to \(-\cos x\), which is also not equivalent to \( \sin x \).
The correct answer is:
**\( f(x) = \cos\left(x - \frac{\pi}{2}\right) \)**