Trigonometric Functions and Their Graphs

Once angles can take any real value, the trigonometric ratios become functions whose graphs reveal their repeating structure. Understanding the shapes of $y=\sin x$, $y=\cos x$, and $y=\tan x$—and how transformations stretch and shift them—lets us model any smoothly oscillating quantity.

The three basic graphs. The graph of $y=\sin x$ is a smooth wave starting at the origin, rising to a maximum of $1$ at $x=\frac{\pi}{2}$, returning to zero at $x=\pi$, dipping to $-1$ at $x=\frac{3\pi}{2}$, and completing one cycle at $x=2\pi$. The graph of $y=\cos x$ has the identical shape but starts at its maximum of $1$ when $x=0$; in fact $\cos x=\sin!\left(x+\frac{\pi}{2}\right)$, so cosine is sine shifted left by $\frac{\pi}{2}$. Both have period $2\pi$, amplitude $1$, and range $[-1,1]$. The graph