Question
Periodic Function Flexibility Quick Check
3 of 53 of 5 Items
Question
Rewrite the cosine function f(x)=cos(13x−π6) as a translated sine function.(1 point)
Responses
f(x)=sin(13x+π)
f left parenthesis x right parenthesis equals sine left parenthesis Start Fraction 1 over 3 End Fraction x plus pi right parenthesis
f(x)=sin(13x−π3)
f left parenthesis x right parenthesis equals sine left parenthesis Start Fraction 1 over 3 End Fraction x plus Start Fraction pi over 3 End Fraction right parenthesis
f(x)=sin(13(x+π))
f left parenthesis x right parenthesis equals sine left parenthesis Start Fraction 1 over 3 End Fraction left parenthesis x plus pi right parenthesis right parenthesis
f(x)=sin(13x)
f left parenthesis x right parenthesis equals sine left parenthesis Start Fraction 1 over 3 End Fraction x right parenthesis
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Answers
GPT-4o mini
To rewrite the function \( f(x) = \cos(13x - \frac{\pi}{6}) \) as a sine function, we can use the identity:
\[
\cos(x) = \sin\left(x + \frac{\pi}{2}\right)
\]
First, let's express \( \cos(13x - \frac{\pi}{6}) \) as a sine function. We can rewrite it as follows:
\[
f(x) = \cos(13x - \frac{\pi}{6}) = \sin\left(13x - \frac{\pi}{6} + \frac{\pi}{2}\right)
\]
Now, we simplify the argument of the sine function:
\[
\frac{\pi}{2} - \frac{\pi}{6} = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}
\]
Thus, we have:
\[
f(x) = \sin\left(13x + \frac{\pi}{3}\right)
\]
So the function \( f(x) = \cos(13x - \frac{\pi}{6}) \) can be rewritten as:
\[
f(x) = \sin(13x + \frac{\pi}{3})
\]
Based on the options given, the correct rewritten form is:
\[
\text{f(x) = sin(13x + } \frac{\pi}{3} \text{)}
\]
Since this exact option is not listed, none of the provided answers accurately reflect the conversion. However, if there's a typo or interpretation, the closest logical interpretation or modified answer would be necessary. Please verify the options available.
\[
\cos(x) = \sin\left(x + \frac{\pi}{2}\right)
\]
First, let's express \( \cos(13x - \frac{\pi}{6}) \) as a sine function. We can rewrite it as follows:
\[
f(x) = \cos(13x - \frac{\pi}{6}) = \sin\left(13x - \frac{\pi}{6} + \frac{\pi}{2}\right)
\]
Now, we simplify the argument of the sine function:
\[
\frac{\pi}{2} - \frac{\pi}{6} = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}
\]
Thus, we have:
\[
f(x) = \sin\left(13x + \frac{\pi}{3}\right)
\]
So the function \( f(x) = \cos(13x - \frac{\pi}{6}) \) can be rewritten as:
\[
f(x) = \sin(13x + \frac{\pi}{3})
\]
Based on the options given, the correct rewritten form is:
\[
\text{f(x) = sin(13x + } \frac{\pi}{3} \text{)}
\]
Since this exact option is not listed, none of the provided answers accurately reflect the conversion. However, if there's a typo or interpretation, the closest logical interpretation or modified answer would be necessary. Please verify the options available.