The graphs of the sine and cosine functions are shown below.
 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 $\,y = 1\,.$
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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 counter-clockwise (start UP). |
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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!
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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:
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As $\,x\,$ goes from $\,0\,$ to $\,\frac{\pi}{2}\,,$ $\,\sin x\,$ goes from $\,0\,$ to $\,1\,.$ (Remember: $\,\frac{\pi}{2}\,$ radians is $\,90^\circ\,$) |
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As $\,x\,$ goes from $\,\frac{\pi}{2}\,$ to $\,\pi\,,$ $\,\sin x\,$ goes from $\,1\,$ back to $\,0\,.$ (Remember: $\,\pi\,$ radians is $\,180^\circ\,$) |
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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:
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domain is $\Bbb R$
(Recall that $\,\Bbb R\,$ represents the set of real numbers.) - range is the interval $[-1,1]$
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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: $$ \cssId{s52}{\cos(x - \frac{\pi}{2}) = \sin x} $$
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Thus, we have two new trigonometric identities!
For all real numbers $\,x\,$:
- $\displaystyle\sin(x + \frac{\pi}{2}) = \cos x$
- $\displaystyle\cos(x - \frac{\pi}{2}) = \sin x$
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Graphing Generalized Sines and Cosines,
like $\,y=a \sin k(x±b)\,$ and $\,y = a \cos(kx±B)$