Graphs of Trig Functions

Sine and cosine — the shape

$y = \sin x$ and $y = \cos x$ both produce smooth waves oscillating between $-1$ and $+1$.

• $\sin x$ starts at the origin: $\sin 0 = 0$.
• $\cos x$ starts at the top: $\cos 0 = 1$.

Both repeat every $2\pi$. That's the period.

Amplitude, period, phase shift

The general form:

$y = A \sin\!\big(B(x - C)\big) + D$

• $|A|$ is the amplitude — height from midline to peak.
• $\dfrac{2\pi}{B}$ is the period — width of one cycle.
• $C$ is the phase shift — horizontal translation.
• $D$ is the vertical shift — moves the midline.

Example. $y = 3 \sin(2x)$ has amplitude $3$ and period $\dfrac{2\pi}{2} = \pi$. It oscillates between $-3$ and $+3$, completing one cycle every $\pi$ units.

Example. $y = \cos\!\left(x - \dfrac{\pi}{2}\right)$ is $\cos$ shifted right by $\pi/2$. At $x = \pi/2$, the input to cosine is $0$, so the output is $1$. The graph looks just like sine — and in fact $\cos\!\left(x - \dfrac{\pi}{2}\right) = \sin x$. Sine and cosine are the same wave, offset by $\pi/2$.

Tangent — vertical asymptotes

$y = \tan x = \dfrac{\sin x}{\cos x}$. It blows up wherever $\cos x = 0$ — at $x = \pm \pi/2, \pm 3\pi/2, \ldots$.

• Period of $\tan$ is $\pi$ (not $2\pi$).
• Range: all real numbers.
• Vertical asymptotes at odd multiples of $\pi/2$.

Key features at a glance

Function	Period	Range	Notes
$\sin x$	$2\pi$	$[-1, 1]$	starts at 0
$\cos x$	$2\pi$	$[-1, 1]$	starts at 1
$\tan x$	$\pi$	$(-\infty, \infty)$	asymptotes at $\pi/2 + k\pi$

Key takeaways

• Sine and cosine are the same wave, offset by $\pi/2$.
• For $A \sin(Bx)$: amplitude $|A|$, period $\dfrac{2\pi}{B}$.
• Tangent has period $\pi$ and asymptotes wherever cosine is zero.