Rewrite the sine function f(d) = -2din(x - pi) +3 as a cosine function. Explain how you got your answer

Answer 1

To rewrite the sine function \( f(d) = -2d \sin(x - \pi) + 3 \) as a cosine function, we can use the relationship between sine and cosine. Specifically, the sine function can be expressed in terms of cosine using the identity:

\[ \sin(x - \pi) = -\cos(x) \]

This is because \( \sin(x - \pi) \) represents the sine of an angle shifted by \( \pi \), which is equivalent to flipping the cosine function.

Steps to Convert the Function:

1. Replace \( \sin(x - \pi) \) with \(-\cos(x)\): \[ f(d) = -2d \sin(x - \pi) + 3 \Rightarrow f(d) = -2d (-\cos(x)) + 3 \]

2. Simplify the equation: \[ f(d) = 2d \cos(x) + 3 \]

Final Result:

So the rewritten function in terms of cosine is:

\[ f(d) = 2d \cos(x) + 3 \]

Explanation:

1. I utilized the sine-cosine identity to convert \( \sin(x - \pi) \) to \( -\cos(x) \).
2. Then, I simplified the expression by multiplying through by the constant \(-2d\) and ensuring the negative signs were correctly accounted for.
3. This transformation maintains the same amplitude and vertical shift in the function, effectively expressing it in terms of cosine as requested.