which of the following describes the relationship between the graphs of $f(x)=cos x$ and $g(x)=sin x$ in the…

which of the following describes the relationship between the graphs of $f(x)=cos x$ and $g(x)=sin x$ in the $xy$-plane?\na the graph of $g$ is a horizontal translation of the graph of $f$.\nb the graph of $g$ is a vertical translation of the graph of $f$.\nc the graph of $g$ is a horizontal dilation of the graph of $f$.\nd the graph of $g$ is a vertical dilation of the graph of $f$.

which of the following describes the relationship between the graphs of $f(x)=cos x$ and $g(x)=sin x$ in the $xy$-plane?\na the graph of $g$ is a horizontal translation of the graph of $f$.\nb the graph of $g$ is a vertical translation of the graph of $f$.\nc the graph of $g$ is a horizontal dilation of the graph of $f$.\nd the graph of $g$ is a vertical dilation of the graph of $f$.

Answer

Explanation:

Step1: Recall trigonometric identity

We know that $\sin x=\cos\left(x - \frac{\pi}{2}\right)$.

Step2: Analyze transformation

The general form of a horizontal translation of a function $y = f(x)$ is $y=f(x - h)$, where $h$ is a non - zero real number. Here, for $f(x)=\cos x$ and $g(x)=\sin x=\cos\left(x-\frac{\pi}{2}\right)$, $h = \frac{\pi}{2}$. This represents a horizontal translation of the graph of $y = \cos x$ to get the graph of $y=\sin x$.

Answer:

A. The graph of $g$ is a horizontal translation of the graph of $f$.