Graphs of Sine and Cosine
The graphs of the sine and cosine functions are shown below.
 Where do these graphs come from?
 Important characteristics of the graphs
 Two trigonometric identities
Graph of $\,y = \sin x$
Graph of $\,y = \cos x$
Where Do These Graphs Come From?
The sine function gives the $y$values of points on the unit circle.
The cosine function gives the $x$values of points on the unit circle.
Since the unit circle has radius $\,1\,,$ all its points have coordinates between $\,1\,$ and $\,1\,.$ That's why both graphs (sine and cosine) are trapped between $\,y = 1\,$ and $\,y = 1\,.$
Visualizing the Graph of the Sine Function
Here's a way you can visualize the graph of the sine function:
Put your finger at the point $\,(1,0)\,$ on the unit circle. Twirl it around the circle counterclockwise (start UP). The sine function tracks your finger's up/down motions:
A Higher Level of Understanding for the Sine Function
In Radian Measure, we ‘wrap’ the real number line around the unit circle. In this way, every real number is associated with a point (called the terminal point) and a corresponding angle on the unit circle. The real number is then the radian measure of this angle!
Wrap the real number line around the unit circle!
Thus, every real number $\,x\,$ ...
... is associated with a point on the unit circle ...
... and a corresponding angle.
In The Unit Circle Approach to Trigonometry, we saw that $\,\sin(x)\,$ is the $y$value of the terminal point:
As $\,x\,$ goes from $\,0\,$ to $\,\frac{\pi}{2}\,,$
$\,\sin x\,$ goes from $\,0\,$ to $\,1\,.$
(Remember: $\,\frac{\pi}{2}\,$ radians is
$\,90^\circ\,$)
As $\,x\,$ goes from $\,\frac{\pi}{2}\,$ to $\,\pi\,,$
$\,\sin x\,$ goes from $\,1\,$ back to $\,0\,.$
(Remember: $\,\pi\,$ radians is
$\,180^\circ\,$)
The Cosine Function
The entire discussion can be repeated to understand the cosine—except that it gives the $x$values of the points, not the $y$values!
Here's how one of the graphics above would be adjusted to focus attention on the $x$value:
Important Characteristics of the Graphs
Properties of both sine and cosine:
Domain and Range
 Domain is $\Bbb R$ (Recall that $\,\Bbb R\,$ represents the set of real numbers.)
 Range is the interval $[1,1]$
Period
 The period is $\,2\pi\,$

Sine has period $\,2\pi\,$: $\,\sin(x+2\pi) = \sin x\,$ for all real numbers $\,x\,.$
More generally: $\,\sin(x+2\pi k) = \sin x\,$ for all integers $\,k\,$ and all real numbers $\,x\,.$

Cosine has period $\,2\pi\,$: $\,\cos(x+2\pi) = \cos x\,$ for all real numbers $\,x\,.$
More generally: $\,\cos(x+2\pi k) = \cos x\,$ for all integers $\,k\,$ and all real numbers $\,x\,.$
Two Trigonometric Identities
Take the sine curve and shift it $\,\frac{\pi}{2}\,$ units to the left—it turns into the cosine curve:
$$ \cssId{s50}{\sin(x + \frac{\pi}{2}) = \cos x} $$Take the cosine curve and shift it $\,\frac{\pi}{2}\,$ units to the right—it turns into the sine curve:
$$ \cssId{s52}{\cos(x  \frac{\pi}{2}) = \sin x} $$Thus, we have two new trigonometric identities! For all real numbers $\,x\,$:
$$ \begin{gather} \cssId{s55}{\sin(x + \frac{\pi}{2}) = \cos x}\cr\cr \cssId{s56}{\cos(x  \frac{\pi}{2}) = \sin x} \end{gather} $$