IB Maths AA 3.7 Notes

Basic sine, cosine, and tangent graphs

Trigonometric graphs show how the values of $\sin x$, $\cos x$ and $\tan x$ change as $x$ changes. These graphs are periodic, which means they repeat in a regular pattern. You also need to understand transformed graphs of the form $f(x)=a\sin(b(x+c))+d$ and similar cosine or tangent forms.

The graph of $y=\sin x$ is a smooth wave.

Its key features are:

• amplitude $1$
• period $2\pi$
• midline $y=0$
• it passes through $(0,0)$

Important points in one cycle are $(0,0)$, $\left(\frac{\pi}{2},1\right)$, $(\pi,0)$, $\left(\frac{3\pi}{2},-1\right)$, $(2\pi,0)$. So the sine graph starts at $0$, rises to $1$, comes back to $0$, falls to $-1$, then returns to $0$.

The graph of $y=\cos x$ is also a smooth wave.

Its key features are:

• amplitude $1$
• period $2\pi$
• midline $y=0$
• it passes through $(0,1)$

Important points in one cycle are $(0,1)$, $\left(\frac{\pi}{2},0\right)$, $(\pi,-1)$, $\left(\frac{3\pi}{2},0\right)$, $(2\pi,1)$. So the cosine graph starts at its maximum value.

The graph of $y=\tan x$ has a different shape.

Its key features are:

• period $\pi$
• undefined amplitude
• vertical asymptotes where $\cos x=0$ or $\frac{\pi}{2} + n\pi$
• it passes through $(0,0)$

The asymptotes occur at $x=\frac{\pi}{2}+k\pi$ for integers $k$. So in one central cycle, the asymptotes are at $x=-\frac{\pi}{2}$ and $x=\frac{\pi}{2}$.

Between these, the graph rises from $-\infty$ to $+\infty$.