Graphing sine, cosine and tangent | Trigonometry

The sine curve

Follow the point on the unit circle as $\theta$ goes from $$0$$ to $2\pi$ and plot its $$y$$ -coordinate. It starts at $$0$$ , rises to $$1$$ at $\frac{\pi}{2}$ , returns to $$0$$ at $\pi$ , falls to $$-1$$ at $\frac{3\pi}{2}$ and returns to $$0$$ at $2\pi$ . Then it repeats forever: that's the sine wave.

RuleKey features of $y = \sin x$ and $y = \cos x$

	$y = \sin x$	$y = \cos x$
domain	all reals	all reals
range	$$[-1, 1]$$	$$[-1, 1]$$
period	$2\pi$	$2\pi$
zeros	$k\pi$	$\frac{\pi}{2} + k\pi$
value at $$x = 0$$	$$0$$	$$1$$ (a maximum)
symmetry	odd (about the origin)	even (about the $$y$$ -axis)

The cosine graph is the sine graph shifted $\frac{\pi}{2}$ to the left.

The tangent graph

Between consecutive asymptotes the graph rises from $-\infty$ to $\infty$ , crossing zero halfway.

Transformations

Sound waves, tides and alternating current are all modeled by functions of the form

y = A\sin\big(B(x - h)\big) + k \qquad \text{or} \qquad y = A\cos\big(B(x - h)\big) + k

with $$B > 0$$ .

Practice

Quick check

What is the period of $y = \sin(4x)$ ?

Quick check

What is the period of $y = \tan x$ ?